  Gauge invariance of systems whose Hamilton-Jacobi equation is separable is
improved by adding surface terms to the action fuctional. The general form of
these terms is given for some complete solutions of the Hamilton-Jacobi
equation. The procedure is applied to the relativistic particle and toy
universes, which are quantized by imposing canonical gauge conditions in the
path integral; in the case of empty models, we first quantize the parametrized
system called ``ideal clock'', and then we examine the possibility of obtaining
the amplitude for the minisuperspaces by matching them with the ideal clock.
The relation existing between the geometrical properties of the constraint
surface and the variables identifying the quantum states in the path integral
is discussed.
