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tokens: in marginally excited dynamos, we restrict ourselves in the following to being real. We now also assume that is constant. In that case, oscillatory solutions were previously thought impossible ( Rädler Bräuer 1987 ). Analogously to their approach, we multiply Eq. ( 8 ) by , where the asterisk denotes complex conjugation, and integrate by parts. Using Eqs. ( 9 ) and ( 10 ), we obtain Furthermore, , so Equation ( 8 ) yields altogether four terms, two of which are real and the other two imaginary. We obtain two integral constraints where denotes the value of on the second boundary at . This implies that (negative frequencies for positive ) and if and . Similar integral constraints can also be formulated for the complex magnetic field, . Unfortunately, the perfect conductor boundary condition, , is more cumbersome. Instead, one could formulate the problem for an artificially modified boundary condition, on . Together with the condition on , the problem for becomes equivalent to that for . In either case, the integral constraints are analogous to those of Rädler Bräuer (1987) ; see Appendix A for details. Fig. 1 Plots of a) real, b) imaginary, and c) absolute parts of . In a) and b) , the zero lines are marked in white, while in c) those of are dotted blue and those of are solid red. 4. The solution Given that Eq. ( 8 ) has constant coefficients, it has solutions proportional to where the index denotes one of two independent solutions. The are in general complex and obey the characteristic equation It has two solutions, To satisfy the boundary conditions ( 9 ) and ( 10 ), we write the solution as a superposition of and . Equation ( 9 ) is readily satisfied by writing where we have ignored the possibility of an arbitrary (complex) constant in front of . To satisfy Eq. ( 10 ), we now require that vanishes. The existence of solutions to is demonstrated by looking at a contour plot of ; see Fig. 1 , where we also plot separately the real and imaginary parts of . We see two zeroes in , which is confirmed by the crossing of the lines where and vanish. (At , there is no such crossing, so is not a solution.) The transcendental equation relating to can be written in more explicit form as To find solutions to , it is convenient to introduce the complex variable We seek solutions to via complex interpolation, where subscripts and refer to the current and previous iteration. This yields the first critical value as with the corresponding complex wavenumbers The wavenumbers and obey the relation which follows from Eqs. ( 9 ) and ( 18 ). The critical values of and were first obtained by Jabbari et al. (2017) using explicit time integration. Additional solutions exist in the second and fourth quadrant of the plane; see Fig. 2 . They are all oscillatory, in agreement with the integral constraints; see Eqs. ( 13 ) and ( 14 ) and Table 1 . However, those higher modes would generally be unstable in a nonlinear calculation and therefore only of limited interest (