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