  This paper aims at the most comprehensive and systematic construction and
tabulation of mechanical systems that admit a second invariant, quadratic in
velocities, other than the Hamiltonian. The configuration space is in general a
2D Riemannian or pseudo-Riemannian manifold and the determination of its
geometry is a part of the process of solution. Forces acting on the system
include a part derived from a scalar potential and a part derived from a vector
potential, associated with terms linear in velocities in the Lagrangian
function of the system. The last cause time-irreversibility of the system. We
construct 41 multi-parameter integrable systems of the type described in the
title mostly on Riemannian manifolds. They are mostly new and cover all
previously known systems as special cases, corresponding to special values of
the parameters. Those include all known cases of motion of a particle in the
plane and all known cases in the dynamics of rigid body. In the last field we
introduce a new integrable case related to Steklov's case of motion of a body
in a liquid. Several new cases of motion in the plane, on the sphere and on the
pseudo-sphere or in the hyperbolic plane are found as special cases.
Prospective applications in mathematics and physics are also pointed out.
