Integrable system on phase space with nonplanar metrics

The problem on the integrability of the evolution system on the phase spaces with the nonplanar metrics is studied. It is shown that in the case, when the phase space is a sphere, the system Hamiltonians are generated under the action of the Poisson operators on the variations of the phase space geodesic lines and the problem on the evolution system integrability is reduced to the task on the integrability of the repers motion equations on the phase space. The bihamiltonian representation of the evaluation systems is connected with the differential-geometric properties of the phase space