  Every second order system of autonomous differential equations can be
described by an autonomous holonomic dynamical system with a Lagrangian part,
an effective potential and a set of generalized forces. The kinematic part of
the Lagrangian defines the kinetic metric which subsequently defines a
Riemannian geometry in the configuration space. We consider the generic
function $I=K_{ab}(t,q^{c})\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q^{c})\dot{q}%
^{a}+K(t,q^{c})$ and require the quadratic first integral condition $dI/dt=0$
without involving any type of symmetry Lie or Noether. Condition $dI/dt=0$
leads to a system of equations involving the coefficients $K_{ab}%
(t,q^{c}),K_{a}(t,q^{c}),K(t,q^{c})$ whose solution will produce all possible
quadratic first integrals of the original system of autonomous differential
equations. We show that the new system of equations relates the quadratic first
integrals of the holonomic system with the geometric collineations of the
kinetic metric and in particular with the Killing tensors of order two. We
consider briefly various results concerning the Killing tensors of second-order
and prove a general formula which gives in the case of a flat kinetic metric
the generic Killing tensor in terms of the vectors of the special projective
algebra of the kinetic metric. This establishes the connection between the
geometry defined by the kinetic metric and the quadratic first integrals of the
original system of differential equations.
