Non-degenerate Singularities of a Three-dimensional Billiard Bounded by an Ellipsoid in a Hooke Potential Field

Consider the problem of motion of a material point under the action of an elastic force of coefficient $$k$$ inside a triaxial ellipsoid whose center coincides with the center of the force field. Such a dynamical system is Liouville integrable in the piecewise-smooth sense. For the attractive and repulsive cases ( $$k>0$$ and $$k<0$$ , respectively), we describe the Liouville foliation of the system in small neighborhoods of regular layers as well as layers containing nondegenerate critical points.