Vladimir Matveev

Consider a Riemannian metric on a surface, and let the geodesic flow of the metric have a second integral that is a third degree polynomial in momenta. Then we can naturally construct a vector field on the surface. We show that the vector field preserves the volume of the surface, and therefore is a Hamiltonian vector field. As examples we treat the Goryachev–Chaplygin top, the Toda lattice and the Calogero–Moser system, and construct their global Hamiltonians. We show that the simpliest choice of Hamiltonian leads to the Toda lattice.

We suggest a simple approach for obtaining integrals of Hamiltonian systems if there is known a trajectorian map of two Hamiltonian systems. An explicite formila is given. As an example, it is proved that if on a manifold are given two Riemannian metrics which are geodesically equivalent then there is a big family of integrals. Our theorem is a generalization of the well-known Painleve–Liouville theorems.

In the present paper we construct and topologically describe a series of examples of metrics on the Klein bottle such that for each metric

• the corresponding geodesic flow has an integral, which is a polynom of degree four in momenta
• the corresponding geodesic flow has no integral, which is a polynom of degree less than four in momenta.

We classify integrable geodesic flows on the torus and on the Klein bottle, which in addition admit a quadratic in momenta first integral. We also study cases if there exists an additional linear integral.

We show that an invariant surface allows to construct the Jacobi vector field along a geodesic line and construct the formula for the normal part of the Jacobi field. If a geodesic line is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic line is hyperbolic) than we can construct the fundamental solution of Jacobi equation $\ddot{u}=-K(t)u$. That was done for quadratically integrable geodesic flows.