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 Liepoint 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.

A conditional symmetry is defined, in the phase-space of a quadratic in velocities constrained action, as a simultaneous conformal symmetry of the supermetric and the superpotential. It is proven that such a symmetry corresponds to a variational (Noether) symmetry.The use of these symmetries as quantum conditions on the wave-function entails a kind of selection rule. As an example, the minisuperspace model ensuing from a reduction of the Einstein -Hilbert action by considering static, spherically symmetric configurations and r as the independent dynamical variable, is canonically quantized. The conditional symmetries of this reduced action are used as supplementary conditions on the wave function. Their integrability conditions dictate, at a first stage, that only one of the three existing symmetries can be consistently imposed. At a second stage one is led to the unique Casimir invariant, which is the product of the remaining two, as the only possible second condition on Ψ. The uniqueness of the dynamical evolution implies the need to identify this quadratic integral of motion to the reparametrisation generator. This can be achieved by fixing a suitable parametrization of the r-lapse function, exploiting the freedom to arbitrarily rescale it. In this particular parametrization the measure is chosen to be the determinant of the supermetric. The solutions to the combined Wheeler -DeWitt and linear conditional symmetry equations are found and seen to depend on the product of the two "scale factors".

We use the conditional symmetry approach to study the r-evolution of a minisuperspace spherically symmetric model both at the classical and quantum level. After integration of the coordinates t, θ and φ in the gravitational plus electromagnetic action the configuration space dependent dynamical variables turn out to correspond to the r-dependent metric functions and the electrostatic field. In the context of the formalism for constrained systems (Dirac -Bergmann, ADM) with respect to the radial coordinate r, we set up a point-like reparameterization invariant Lagrangian. It is seen that, in the constant potential parametrization of the lapse, the corresponding minisuperspace is a Lorentzian three-dimensional flat manifold which obviously admits six Killing vector fields plus a homothetic one. The weakly vanishing r-Hamiltonian guarantees that the phase space quantities associated to the six Killing fields are linear holonomic integrals of motion. The homothetic field provides one more rheonomic integral of motion. These seven integrals are shown to comprise the entire classical solution space, i.e. the space-time of a Reissner-Nordström black hole, the rreparametrization invariance since one dependent variable remains unfixed, and the two quadratic relations satisfied by the integration constants. We then quantize the model using the quantum analogues of the classical conditional symmetries, and show that the existence of such symmetries yields solutions to the Wheeler-DeWitt equation which, as a semiclassical analysis shows, exhibit a good correlation with the classical regime. We use the resulting wave functions to investigate the possibility of removing the classical singularities.