  The investigation of contact symmetries of re--parametrization invariant
Lagrangians of finite degrees of freedom and quadratic in the velocities is
presented. The main concern of the paper is those symmetry generators which
depend linearly in the velocities. A natural extension of the symmetry
generator along the lapse function $N(t)$, with the appropriate extension of
the dependence in $\dot{N}(t)$ of the gauge function, is assumed; this action
yields new results. The central finding is that the integrals of motion are
either linear or quadratic in velocities and are generated, respectively by the
conformal Killing vector fields and the conformal Killing tensors of the
configuration space metric deduced from the kinetic part of the Lagrangian
(with appropriate conformal factors). The freedom of re--parametrization allows
one to appropriately scale $N(t)$, so that the potential becomes constant; in
this case the integrals of motion can be constructed from the Killing fields
and Killing tensors of the scaled metric. A rather interesting result is the
non--necessity of the gauge function in Noether's theorem due to the presence
of the Hamiltonian constraint
