Vector fields and differential equations

The geometry of orbits of families of smooth vector fields is an important object of mathematics due to its importance in applications, in the theory of dynamic systems and in the foliation theory. The paper devoted to the applications of the geometry of orbits of vector fields in four dimensional Euclidean space in theory of differential equations. It is shown that orbits generate singular foliation ever regular leaf of which is a surface of negative Gauss curvature and zero normal torsion. In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields. In this paper smoothness is smoothness of the class C ∞.