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the geodesics in a kerr metric are considered in the classical books on general relativity [ 13 ] . some recent papers are devoted to more detailed study of geodesics on kerr s black hole with the aim to elucidate the mechanism of jet formation [ 4 ] , and to analyze the possibility of particle acceleration to arbitrary high energy [ 5 ] . the complete sets of analytic solutions of the geodesic equation in axially symmetric space - time are given in [ 6 ] . however , the description of particle motion by geodesics is restricted to a spinless particle . the motion of a spinning test particle is described by the mathisson - papapetrou - dixon equations [ 79 ] : @xmath2 @xmath3 where @xmath4 is the particle s 4-velocity , @xmath5 is the tensor of spin , @xmath6 and @xmath7 are , respectively , the mass and the covariant derivative with respect to the particle s proper time @xmath8 ; @xmath9 is the riemann curvature tensor ( units @xmath10 are used ) . it is necessary to add a supplementary condition to eqs . ( 1 ) , ( 2 ) in order to choose an appropriate trajectory of the particle s center of mass . most often the conditions @xmath11 and @xmath12 are used , where @xmath13 is the 4-momentum . in practice , the condition for a spinning test particle @xmath14 must be taken into account [ 10 ] , where @xmath15 is the absolute value of spin , @xmath16 is the coordinate distance of the particle from the massive body . after [ 79 ] , eqs . ( 1 ) , ( 2 ) were obtained in many papers by different approaches [ 11 ] . also , this subject is of importance in some recent publications [ 12 ] .    in general , the solutions of eqs . ( 1 ) , ( 2 ) under conditions ( 3 ) and ( 4 ) are different . however , in the post - newtonian approximation these solutions coincide with high accuracy [ 13 ] , just as in the case if the spin effects can be described by a convergent in spin series , as some corrections to the corresponding geodesic expressions [ 14 ] . therefore , instead of rigorous mpd eqs . ( 1 ) their linear spin approximation @xmath17 is often considered . in this approximation condition ( 4 ) coincides with ( 3 ) . the effect of spin on the particle s motion in kerr s field has been studied since the 1970s [ 10,15,16 ] . in the past 1012 years this subject gives rise to renewed interest [ 1722 ] , particularly in the context of investigations of the possible chaotic motions [ 17,19 ] . also , these references provide a good introduction concerning the mpd equations . the purpose of this paper is to investigate more carefully the world lines and trajectories of a spinning particle moving relative to a kerr source with the velocity close to the velocity of light . we focus on the circular and close to circular orbits , because just on these orbits one can expect the significant effects of the spin - gravity interaction [ 10,2325 ] . indeed , these orbits are of interest in the context of investigations of the nongeodesic synchrotron electromagnetic radiation of highly relativistic protons and electrons near black holes . besides , it is known that the highly relativistic circular orbits of a spinless particle are of importance in the classification of all possible geodesic orbits in a kerr spacetime . naturally , the circular highly relativistic orbits of a spinning particle are of importance in the classification of all possible significantly nongeodesic orbits in this spacetime as well . also , these orbits are exclusive in the sense that they are described by the strict analytical solutions of the mpd equations . the main features of the spin - gravity interaction that are revealed on the circular and close to circular orbits will be a good reference to further investigations of most general motions of a spinning particle in a kerr spacetime .    we stress that mpd equations are the classical limit of the general relativistic dirac equation [ 26 ] , new results in this context are presented in [ 27 ] . therefore , highly relativistic solutions of the mpd equations stimulate the corresponding investigations of the fermion s interaction with the strong gravitational field . the paper is organized as follows . sec . 2 deals with the relationships following from the mpd equations for the highly relativistic equatorial circular orbits of a spinning particle in kerr s field , in the boyer - lindquist coordinates . the linear spin mpd equations for any motions of a spinning particle are considered in sec . the results of computer integration of these equations for some significantly nongeodesic motions are presented in sec . 4 . we conclude in sec . 5 .    following [ 3 ] , in this paper @xmath0 notes the radial coordinate of the photon circular orbits in the case of the counter - rotation . in practical calculations it is convenient to represent the mpd equations through the spin 3-vector @xmath18 , instead of the 4-tensor @xmath5 , where by definition @xmath19 where @xmath20 is the determinant of @xmath21 , @xmath22 is the levi - civita symbol ( here and in the following , latin indices run 1 , 2 , 3 , and greek indices 1 , 2 , 3 , 4 , unless otherwise specified ) . ( 2 ) can be written as [ 23 ]    @xmath23 } - u^\pi u_\rho \gamma^\rho_{\pi[4 } u_{i]})s_k u^k\ ] ] @xmath24 } u^\pi = 0,\ ] ] where a dot denotes differentiation with respect to the proper time @xmath8 , and square brackets denote antisymmetrization of indices ; @xmath25 are the christoffel symbols . ( 7 ) in terms of @xmath18 is @xmath26 @xmath27    in many papers the 4-vector of spin @xmath28 is considered , where @xmath29 the following relationship holds : @xmath30    let us consider eqs . ( 9 ) , ( 10 ) for the kerr metric using the boyer - lindquist coordinates @xmath31 then the nonzero components of @xmath21 are @xmath32 @xmath33 @xmath34 where @xmath35 [ in the following we shall put @xmath36 , without any loss in generality ; the metric signature is    + . ] it is easy to check that three equations from ( 9 ) have a partial solution with @xmath37 , @xmath38 , @xmath39 and the relationship for the nonzero component of the spin 3-vector @xmath40 is @xmath41 where @xmath42 is the constant of integration . the physical meaning of this constant is the same as in the general integral of the mpd equations [ 15 ] @xmath43 we stress that relationship ( 12 ) is valid for any equatorial motions ( @xmath37 ) , with the spin orthogonal to the motion plane ( @xmath44 ) . the possible equatorial orbits of a spinning particle are described by eq . first , we shall consider the case of the circular orbits with @xmath45 investigating the conditions of existence of the equatorial circular orbits for a spinning particle in kerr s field we use eqs . ( 10 ) , ( 12 ) , and @xmath46 . it is known that from the geodesic equations in this field , the algebraic relationship that follows determines the dependence of the velocity of a spinless particle on the radial coordinate @xmath16 of the equatorial circular orbit . similarly , from the first equation of set ( 10 ) using eq . ( 14 ) we obtain the relationship for the equatorial circular orbits of a spinning particle in kerr s field as follows : @xmath47 @xmath48=0.\ ] ] [ by eq . ( 14 ) other equations of set ( 10 ) are automatically satisfied . ] taking into account eq . ( 12 ) and the explicit expressions for @xmath49 and @xmath9 ( see , e.g. , [ 18 ] ) from eq . ( 15 ) we obtain @xmath50 @xmath51=0.\ ] ] the 4-velocity component @xmath52 can be expressed through @xmath53 from the condition @xmath46 as follows : @xmath54 ( just the sign `` + '' at the radical in eq . ( 17 ) ensures the positive value of @xmath52 ) . inserting expression ( 17 ) into eq . ( 16 ) and eliminating the radical by raising to the second power we get @xmath55 + 12\varepsilon mar^4\delta-9\varepsilon^2m^2r^4\delta)\ ] ] @xmath56 @xmath57 where , as in eq . ( 6 ) , @xmath58 . without any loss in generality we put @xmath59 , then by eq . ( 12 ) @xmath60 . so , the particle s angular velocity @xmath61 on the circular orbit with the radial coordinate @xmath16 must satisfy eq . ( let us show that eq . ( 18 ) provides known solutions . in the partial case of a spinless particle ( @xmath62 ) from eq . ( 18 ) we have @xmath63}.\ ] ] it follows from eq . ( 19 ) that the velocity of such a particle on the circular orbit is highly relativistic if the expression @xmath64 is close to @xmath65 . this fact is known from the analysis of the geodesic orbits in a kerr field , as well as that just the roots of the equation @xmath66 determine the values of @xmath16 for the photon orbits .    if @xmath67 and the absolute value of the expression @xmath68 $ ] in the factor of @xmath69 in eq . ( 18 ) is much greater than @xmath70 , then it is easy to verify that the corresponding roots of eq . ( 18 ) describe the circular orbits with the angular velocity which is close to the angular velocity of the corresponding geodesic orbits due to @xmath71 . more exactly , in this case we have @xmath72 that is not described in the literature . namely , it is not difficult to calculate that for @xmath67 eqs . ( 16 ) , ( 18 ) have the solutions which describe the highly relativistic circular orbits with the values of @xmath16 that is equal or close to @xmath0 , i.e. , to the radial coordinate of the counter - rotation photon circular orbits . for example , in the case of the maximum kerr field ( @xmath73 ) the orbits with @xmath74 , where @xmath75 , are highly relativistic , both for positive and negative @xmath76 . if @xmath77 , according to eqs . ( 16 ) , ( 18 ) the values of @xmath53 on these orbits are determined by the expression @xmath78 [ the choice of the sign in eq . ( 21 ) , @xmath79 , is dictated by the necessity to satisfy both eq . ( 18 ) and ( 16 ) ; eq . ( 18 ) , as compared to ( 16 ) , has additional roots because of the operation of raising to the second power . ] similarly as in the case of eq . ( 20 ) , it is easy to check that if @xmath80 it follows from eqs . ( 16 ) , ( 18 ) the expression for @xmath53 which in the main term coincides with the known analytic solution for the corresponding geodesic circular orbit . it follows from eqs . ( 16 ) , ( 18 ) at @xmath1 for any @xmath81 that @xmath82 it is known from the geodesic equations that the values of @xmath0 increase monotonically from @xmath83 at @xmath84 to @xmath85 at @xmath73 . thus , according to eq . ( 22 ) the expression for the angular velocity @xmath61 in the main term is proportional to @xmath86 , whereas the angular velocity in eq . ( 20 ) at @xmath87 , @xmath88 is proportional to @xmath89 . further details appear below in this sec .    using eq . ( 22 ) we can estimate the value of the lorentz @xmath90-factor , corresponding to the 4-velocity component @xmath53 , for different @xmath91 . more exactly , we shall calculate the lorentz @xmath90-factor from the point of view of an observer which is at rest relative to a kerr mass . according to the general expression for the 3-velocity components @xmath92 we write [ 1 ] @xmath93 and for the second power of the velocity absolute value @xmath94 we have @xmath95 where @xmath96 is the 3-space metric tensor . the relationship between @xmath96 and @xmath21 is as follows : @xmath97 for the circular motions we have @xmath98 , and according to eq . ( 23 ) @xmath99 by eqs . ( 24)-(26 ) and the condition @xmath46 , for the @xmath90-factor we write @xmath100 inserting the value @xmath53 from eq . ( 22 ) into eq . ( 27 ) we find in the corresponding spin approximation @xmath101 it follows from eq . ( 28 ) that @xmath102 , i.e. , the under consideration circular motions are highly relativistic . fig . 1 shows the dependence @xmath103 on @xmath104 , where @xmath105 is the @xmath90-factor for @xmath106 , that is , at @xmath84 . ratio @xmath90-factor at different @xmath91 , from @xmath107 , to @xmath90-factor at @xmath84 vs. radial coordinate.,width=264 ]    it is known that the important characteristic of the particle s motion in the kerr spacetime are its energy and angular momentum . let us estimate the conserved values of the energy @xmath108 and angular momentum @xmath109 of a spinning particle on the above considered highly relativistic circular orbits with eqs . ( 21 ) , ( 22 ) . the expressions for these quantities are [ 16,28 ] @xmath110 @xmath111 by eqs . ( 21 ) , ( 29 ) for @xmath74 , @xmath77 we obtain @xmath112 the energy of a spinless particle on the geodesic circular orbit with @xmath74 , @xmath113 for @xmath73 is equal to @xmath114 it follows from eqs . ( 31 ) , ( 32 ) that @xmath115 at the same time according to ( 31 ) , ( 32 ) for @xmath116 we have @xmath117 that is , the values of energy of the spinning and spinless particles on the highly relativistic circular orbits with the same @xmath74 in the maximal kerr field can differ significantly . it is easy to show that similar situation takes place for all values @xmath81 with @xmath87 . in addition , one can estimate for these circular orbits that according to eq . ( 30 ) @xmath118 .    as a result , the relationships @xmath119 and @xmath118 following from eqs . ( 29 ) , ( 30 ) show clearly that the corresponding solutions of the mpd equations can not be obtained in the framework of the analytic perturbation approach to the dynamics of a classical spinning particle developed in [ 2022 ] . we stress that in this sec . above we have considered the new highly relativistic circular solutions of the approximate mpd eqs . ( 2 ) , ( 7 ) . let us show that these solutions satisfy the rigorous mpd eqs . ( 1 ) , ( 2 ) . indeed , it follows from eqs . ( 1 ) that their terms , which were neglected in eqs . ( 7 ) , in the case of the circular equatorial motions are presented in the first eq . of set ( 1 ) only . in metric ( 11 ) these terms can be written as @xmath120 where @xmath121 @xmath122 @xmath123,\ ] ] @xmath124 @xmath125 first , we point out that according to eq . ( 36 ) in the case of a schwarzschild s field for the circular orbit with @xmath126 the all coefficients @xmath127 are equal to 0 . therefore , in this case expression ( 35 ) is equal to 0 identically , independently on the explicit expressions for @xmath53 , @xmath52 . it means that the highly relativistic circular orbit of a spinning particle with @xmath126 in a schwarzschild s field is a common strict solution both of the approximate mpd equations ( 2 ) , ( 7 ) and the rigorous mpd equations ( 1 ) , ( 2 ) . second , it is not difficult to check that applying eqs . ( 17 ) , ( 22 ) to expression ( 35 ) at @xmath128 yields @xmath65 in the main terms , i.e. , within the accuracy of order @xmath129 . thus , in this sec . we dealt with new partial solutions of the mpd equations in a kerr spacetime with eqs . ( 12 ) , ( 14 ) , ( 22 ) . all highly relativistic orbits of a spinning particle described by these solutions are circular and located in the space region with @xmath87 , @xmath75 . to study highly relativistic orbits beyond this region it is necessary to carry out the corresponding computer calculations . it is an aim of secs . 3 and 4 . now the point of interesting is the noncircular highly relativistic motions of a spinning particle that starts near @xmath0 in a kerr field . in particular , we shall consider the effect of the 3-vector @xmath18 inclination to the equatorial plane @xmath37 on the particle s trajectory . in this case ( 9 ) can not be integrated separately from eqs . ( 7 ) . for computer integration , it is necessary to write the explicit form of eqs . ( 7 ) , ( 9 ) in metric ( 11 ) . it is convenient to use the dimensionless quantities @xmath130 , where by definition @xmath131 @xmath132 @xmath133 and @xmath134 ( in contrast to @xmath129 from ( 6 ) , that depends on @xmath16 , here @xmath135 is defined to be @xmath136 ) . then it follows from eqs . ( 7 ) , ( 9 ) the set of 11 first - order differential equations @xmath137 @xmath138 @xmath139 where a dot denotes differentiation with respect to @xmath140 , and @xmath141 ( @xmath142 ) are some functions which depend on @xmath143 . because the expressions for @xmath144 in general case of any @xmath91 are too lengthy , here we write @xmath144 for the much simpler case @xmath84 : @xmath145 @xmath146 @xmath147 @xmath148 @xmath149 @xmath150 @xmath151 @xmath152 @xmath153 @xmath154+\frac{y_9}{y_8}a_4,\ ] ] @xmath155 @xmath156 @xmath157 @xmath158,\ ] ] @xmath159 @xmath160 @xmath161 @xmath162 @xmath163+\frac{y_{11}}{y_8}a_4,\ ] ] where @xmath164 according to eqs . ( 13 ) , ( 38 ) the expression for @xmath165 can be written as @xmath166 @xmath167 @xmath168 eqs . ( 39)(41 ) are aimed at computer integration . we present here the results of computer integration of eqs . ( 39 ) with ( 40 ) , ( 41 ) . all plots below are restricted to the domains of validity of the linear spin approximation . that is , in these domains the neglected terms of the rigorous mpd equations are much less than the linear spin terms . we monitor errors of computing using the conserved quantities : the absolute value of spin , the energy and angular momentum ( see eqs . ( 13 ) , ( 29 ) , ( 30 ) ) . 29 correspond to the case of schwarzschild s field . all plots start with the same initial values of the coordinates @xmath169 , namely , at @xmath170 and @xmath65 correspondingly . we do not vary the initial values of the 4-velocity components @xmath171 and @xmath53 as well . more exactly , we put @xmath172 and @xmath173 where expression ( 42 ) is the solution of eq . ( 18 ) at @xmath84 , rigorous in @xmath135 . that is , the initial values of @xmath171 and @xmath53 are the same as for the equatorial circular orbit with @xmath126 ( it is easy to check that expression ( 42 ) coincides with ( 22 ) in the corresponding spin approximation ) . however , we vary the initial inclination angle of the spin 3-vector to the plane @xmath174 , without change of the absolute value of spin ( figs . 26 ) , and add the small perturbation by the radial velocity ( figs . 79 ) . without any loss in generality we put @xmath175 . for comparison , we present the corresponding solutions of the geodesic equations that start with the same initial values of the coordinates and velocity as the solutions of the equations for a spinning particle . in all cases as a typical value we put @xmath176 .    figs . 29 let us compare the world lines and trajectories of the spinning and spinless particles in schwarzschild s field . 2 , 6 , 7 , 9 exhibit the significant repulsive effects of the spin - gravity interaction on the spinning particle . due to the repulsive action the spinning particle falls on the horizon surface during longer time as compared to the spinless particle ( fig . 2 ) . moreover , according to figs . 6 , 9 the considerable space separation takes place within a short time , i.e. , within the time of the spinless particle s fall on the horizon . the point of interest is the phenomenon when a spinning particle orbits below the equatorial plane for some revolutions , figs . 3 , 8 ( we recall that according to the geodesic equations similar situation is impossible for a spinless particle ) . it is easy to calculate that for @xmath177 a spinning particle can orbit above the equatorial plane . we stress that all graphs in figs . 29 are interrupted beyond the domain of validity of the linear spin approximation . by figs . 3 , 4 , within the time of this approximation validity , the period of the @xmath178-oscillation coincides with the period of the particle s revolution by @xmath179 . whereas on this interval the value of the spin component @xmath180 is @xmath136 ( fig . 5 ) , just as the components @xmath181 and @xmath182 ( the corresponding graphs are not presented here ) . we point out that this situation differs from the corresponding case of the circular motions of a spinning particle that are not highly relativistic . indeed , then the nonzero radial spin component is not @xmath136 but oscillates with the period of the particle s revolution by @xmath179 . besides , in the last case the mean level of @xmath178 coincides with @xmath183 , whereas in figs . 3 , 8 the mean values of @xmath178 are above @xmath183 . according to figs . 3 , 8 the amplitude of the @xmath178-oscillation increases with the inclination angle . however , @xmath184 is small even for the inclination angle that is equal to @xmath183 .    in the context of figs . 3 , 8 we point out an interesting analytical result following from eqs . ( 39)(41 ) . namely , it is not difficult to check that these equations are satisfied if @xmath185 @xmath186 @xmath187 @xmath188 @xmath189 where @xmath190 and @xmath191 are some small constant values such that @xmath192 , @xmath193 . according to eq . ( 41 ) the relationship between @xmath190 , @xmath191 and @xmath129 from eq . ( 6 ) is as follows : @xmath194 by notation ( 37 ) we conclude that the partial solution of eqs . ( 39)(40 ) , that is presented in eq . ( 43 ) , describes the highly relativistic nonequatorial circular motion with @xmath195 . that is , eq . ( 43 ) shows the possibility of the spinning particle solution that orbit permanently above or below the equatorial plane , however , with the small value @xmath196 only . one can verify that due to the small values @xmath190 , @xmath191 solution ( 43 ) is an approximate solution of the rigorous mpd equations .    just as in the case of schwarzschild s field , the set of eqs . ( 39 ) with the corresponding expressions for @xmath144 can be integrated numerically in kerr s field . 10 , as an analogy of fig . 2 , shows the plots of @xmath197 for some values of the inclination angle at @xmath73 . 11 shows the dependence of the amplitude and period of the @xmath178-oscillation on the kerr parameter @xmath91 . the main features of figs . 49 are peculiar to the corresponding plots for kerr s field , that are not presented here for brevity .    according to figs . 2 , 6 , 10 when the inclination angle is equal to @xmath183 , i.e. , when spin lies in the equatorial plane , so that spin - gravity coupling is equal to @xmath65 , the corresponding plots coincide with the geodesics . it is an evidence of the correct transition from solutions of the mpd equations to geodesics .    fig . 12 illustrates some typical cases of the equatorial motions of a particle with fixed initial values of its coordinates and velocity but with different absolute value of spin ( the inclination angle is equal to @xmath198 ) in kerr s field at @xmath73 . all curves start from @xmath199 with zero radial velocity and the tangential velocity which at @xmath176 is needed for the circular motion with @xmath199 . this circular motion is shown by the horizontal line , whereas other curves represent the noncircular motions at @xmath200 with the same particle s initial values of coordinates and velocity . according to fig . 12 the circular orbit of a spinning particle with @xmath199 monotonically trend to the corresponding noncircular geodesic orbit if @xmath135 trend to 0 , i.e. , the limiting transition @xmath201 is correct . radial coordinate vs. proper time for the inclination angle @xmath198 ( horizontal line @xmath126 ) , @xmath202 ( dash and dot line ) , @xmath203 ( dash line ) , and @xmath183 ( solid line ) at @xmath84 , @xmath176 . the dot line corresponds to the geodesic motion with the same initial values of the coordinates and velocity.,width=264 ]     graphs of the angle @xmath178 vs. proper time for the inclination angle @xmath198 ( horizontal line @xmath204 , @xmath202 ( solid line ) , @xmath203 ( dash line ) , and @xmath183 ( dash and dot line ) at @xmath84 , @xmath176.,width=249 ]     graphs of the angle @xmath179 vs. proper time at @xmath84 , @xmath176 for different values of the inclination angle practically coincide with the corresponding geodesic plot . the same feature takes place for the corresponding graphs t vs. s that are not presented here for brevity.,width=283 ]     graphs of @xmath180 vs. proper time for the inclination angle @xmath202 ( dash line ) , @xmath203 ( dash and dot line ) , and @xmath183 ( solid line ) at @xmath84 , @xmath176.,width=283 ]     trajectories of the spinning particle in the polar coordinates for the inclination angle @xmath198 ( circle @xmath126 ) , @xmath202 ( dash line ) , @xmath203 ( dash and dot line ) , and @xmath183 ( solid line ) at @xmath84 , @xmath176 . the dot line corresponds to the geodesic motion with the same initial values of the coordinates and velocity . the circle @xmath205 corresponds to the horizon line.,width=226 ]     radial coordinate vs. proper time for the inclination angle @xmath198 ( dash and dot line ) , @xmath202 ( dash line ) , and @xmath203 ( solid line ) at @xmath84 , @xmath176 , and the nonzero radial velocity @xmath206 . the dot line corresponds to the geodesic motion with the same initial values of the coordinates and velocity.,width=264 ]     graphs of the angle @xmath178 vs. proper time for the inclination angle @xmath198 ( horizontal line ) , @xmath202 ( dash line ) and @xmath203 ( solid line ) , at @xmath84 , @xmath176 , and @xmath206.,width=264 ]     trajectories of the spinning particle in the polar coordinates for the inclination angle @xmath198 ( dash and dot line ) , @xmath202 ( dash line ) , and @xmath203 ( solid line ) at @xmath84 , @xmath176 , and @xmath207 . the dot line corresponds to the geodesic motion with the same initial values of the coordinates and velocity.,width=226 ]     radial coordinate vs. proper time for the inclination angle @xmath198 ( horizontal line @xmath199 ) , @xmath202 ( dash and dot line ) , @xmath203 ( dash line ) , and @xmath183 ( solid line ) at @xmath73 , @xmath176 . the dot line corresponds to the geodesic motion with the same initial values of the coordinates and velocity . , width=264 ]     graphs of the angle @xmath178 vs. proper time for the inclination angle @xmath202 at @xmath84 ( dash line ) , @xmath208 ( dash and dot line ) , @xmath73 ( solid line ) ; @xmath176.,width=264 ]     radial coordinate vs. proper time for the equatorial motions in kerr s field at @xmath73 with different absolute value of spin and the same ( common ) initial values of the coordinates and velocity . the circular orbit with @xmath199 at @xmath176 is shown by the long dash line . the solid , dash and dot , and dash lines describe the cases when @xmath135 is equal to @xmath209 , @xmath210 , and @xmath211 correspondingly . the dot line represents the geodesic motion with the same initial values of the coordinates and velocity.,width=226 ]    finally , we remark on a simple conclusion following from the nongeodesic curves of a spinning particle presented , in part , in figs . 212 . let us consider any point on the trajectory of a spinning particle corresponding to its proper time @xmath212 ( we recall that all curves in figs . 212 begin at @xmath213 ) . then the geodesic curve can be calculated which starts just at this point with the velocity that is equal to the velocity of the spinning particle at the same point . also , it is not difficult to estimate the deviation of the pointed out nongeodesic curve from this geodesic . in principle , it means that in our comparison of the corresponding geodesic and nongeodesic curves we are not restricted to the trajectories of a spinning particle which start in the small neighborhood of the value @xmath1 only ( naturally , here we are restricted to the domain of validity of the linear spin approximation ) . this conclusion may be useful for the generalization of the results obtained in this sec . on other motions of a spinning particle . in this paper , using the linear spin approximation of the mpd equations , we have studied the significantly nongeodesic highly relativistic motions of a spinning particles starting near @xmath1 in kerr s field . some of these motions , namely circular , are described by the analytical relationships following directly from mpd equations in the boyer - lindquist coordinates . others , noncircular and nonequatorial , are calculated numerically . for realization of these motions the spinning particle must possess the obital velocity corresponding to the relativistic lorentz @xmath90-factor of order @xmath86 . all considered cases of the spinning particle motion are within the framework of validity of the test - particle approximation when @xmath214 . the situation with a macroscopic test particle moving relative to a massive body with @xmath215 is not realistic . however , the highly relativistic values of the lorentz @xmath90-factor are usual in astrophysics for the elementary particles . for example , if @xmath216 is equal to three of the sun s mass ( as for a black hole ) , then @xmath135 for an electron is of order @xmath217 and from eq . 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using the mathisson - papapetrou - dixon ( mpd ) equations , we investigate the trajectories of a spinning particle starting near @xmath0 in a kerr field and moving with the velocity close to the velocity of light ( @xmath0 is the boyer - lindquist radial coordinate of the counter - rotation circular photon orbits ) . 
 first , as a partial case of these trajectories , we consider the equatorial circular orbit with @xmath1 . 
 this orbit is described by the solution that is common for the rigorous mpd equations and their linear spin approximation . 
 then different cases of the nonequatorial motions are computed and illustrated by the typical figures . 
 all these orbits exhibit the effects of the significant gravitational repulsion that are caused by the spin - gravity interaction . 
 possible applications in astrophysics are discussed .