New classes of quadratically integrable systems with velocity dependent potentials: non-subgroup type cases

We study quadratic integrability of systems with velocity dependent potentials in three-dimensional Euclidean space. Unlike the case only scalar potential, does not imply separability configuration The leading order terms pairs commuting integrals can either generalize or have no relation to forms separation absence a vector potential. call such generalized, distinguish them from standard ones, which would correspond separation. Here we focus on three cases generalized non-subgroup type integrals, namely elliptic cylindrical, prolate/oblate spheroidal and circular parabolic together one related any coordinate system. find two new integrable systems, non-separable space, both cylindrical integrals. In other cases, all found were already known possess limit vanishing reduce free motion therefore separate every orthogonal