  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.
