  We consider the application of the theory of symmetries of coupled ordinary
differential equations to the case of reparametrisation invariant Lagrangians
quadratic in the velocities; such Lagrangians encompass all minisuperspace
models. We find that, in order to acquire the maximum number of symmetry
generators, one must (a) consider the lapse $N(t)$ among the degrees of freedom
and (b) allow the action of the generator on the Lagrangian and/or the
equations of motion to produce a multiple of the constraint, rather than
strictly zero. The result of this necessary modification of the standard theory
(concerning regular systems) is that the Lie - point symmetries of the
equations of motion are exactly the variational symmetries (containing the time
reparametrisation symmetry) plus the well known scaling symmetry. These
variational symmetries are seen to be the simultaneous conformal Killing fields
of both the metric and the potential, thus coinciding with the conditional
symmetries defined in phase space. In a parametrisation of the lapse for which
the potential becomes constant, the generators of the aforementioned symmetries
become the Killing fields of the scaled supermetric and the homothetic field
respectively.
