New examples of conservative systems on S^2 possessing an integral cubic in momenta

It has been proved that on 2-dimensional orientable compact manifolds of genus $g>1$ there is no integrable geodesic flow with an integral polynomial in momenta. There is a conjecture that all integrable geodesic flows on $T^2$ possess an integral quadratic in momenta. All geodesic flows on $S^2$ and $T^2$ possessing integrals linear and quadratic in momenta have been described by Kolokol'tsov, Babenko and Nekhoroshev. So far there has been known only one example of conservative system on $S^2$ possessing an integral cubic in momenta: the case of Goryachev-Chaplygin in the dynamics of a rigid body. The aim of this paper is to propose a new one-parameter family of examples of complete integrable conservative systems on $S^2$ possessing an integral cubic in momenta. We show that our family does not include the case of Goryachev-Chaplygin.