On the geometry of the orbits of Killing vector fields

Let \(D\) be a set of smooth vector fields on the smooth manifold \(M\).It is known that orbits of \(D\) are submanifolds of M. Partition \(F\) of M into orbits of \(D\) is a singular foliation. In this paper we are studying geometry of foliation which is generated by orbits of a family of Killing vector fields.In the case \(M=R^3\) it is obtained full geometrical classification of \(F\). Throughout this paper the word "smooth" refers to a class \(C^\infty\).