On Algebraic Integrals of the Motion of Point over a Quadric in Quadratic Potential

The motion of a point over an $n$-dimensional nondegenerate quadric in one-dimentioned quadratic potential under the assumption that there exist $n+1$ mutually orthogonal planes of symmetry is considered. It is established, that all cases of the existence of an algebraic complete commutative set of integrals are exhausted by classical ones. The question whether the integrability due to Liouville is inherited by invariant symplectic submanifolds is studied. In algebraic category for submanifolds of dimension $4$ such integrability is valid.