  This is a continuation of the work initiated in a previous paper on so-called
driven cofactor systems, which are partially decoupling second-order
differential equations of a special kind. The main purpose in that paper was to
obtain an intrinsic, geometrical characterization of such systems, and to
explain the basic underlying concepts in a brief note. In the present paper we
address the more intricate part of the theory. It involves in the first place
understanding all details of an algorithmic construction of quadratic integrals
and their involutivity. It secondly requires explaining the subtle way in which
suitably constructed canonical transformations reduce the Hamilton-Jacobi
problem of the (a priori time-dependent) driven part of the system into that of
an equivalent autonomous system of St\"ackel type.
