  We consider the generic quadratic first integral (QFI) of the form
$I=K_{ab}(t,q)\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q)\dot{q}^{a}+K(t,q)$ and require
the condition $dI/dt=0$. The latter results in a system of partial differential
equations which involve the tensors $K_{ab}(t,q)$, $K_{a}(t,q)$, $K(t,q)$ and
the dynamical quantities of the dynamical equations. These equations divide in
two sets. The first set involves only geometric quantities of the configuration
space and the second set contains the interaction of these quantities with the
dynamical fields. A theorem is presented which provides a systematic solution
of the system of equations in terms of the collineations of the kinetic metric
in the configuration space. This solution being geometric and covariant,
applies to higher dimensions and curved spaces. The results are applied to the
simple but interesting case of two-dimensional (2d) autonomous conservative
Newtonian potentials. It is found that there are two classes of 2d integrable
potentials and that superintegrable potentials exist in both classes. We
recover most main previous results, which have been obtained by various
methods, in a single and systematic way.
