  A method of quantizing parametrized systems is developed that is based on a
kind of ``gauge invariant'' quantities---the so-called perennials (a perennial
must also be an ``integral of motion''). The problem of time in its particular
form (frozen time formalism, global problem of time, multiple choice problem)
is met, as well as the related difficulty characteristic for this type of
theory: the paucity of perennials. The present paper is an attempt to find some
remedy in the ideas on ``forms of relativistic dynamics'' by Dirac. Some
aspects of Dirac's theory are generalized to all finite-dimensional first-class
parametrized systems. The generalization is based on replacing the Poicar\'{e}
group and the algebra of its generators as used by Dirac by a canonical group
of symmetries and by an algebra of elementary perennials. A number of insights
is gained; the following are the main results. First, conditions are revealed
under which the time evolution of the ordinary quantum mechanics, or a
generalization of it, can be constructed. The construction uses a kind of gauge
and time choice and it is described in detail. Second, the theory is structured
so that the quantum mechanics resulting from different choices of gauge and
time are compatible. Third, a practical way is presented of how a broad class
of problems can be solved without the knowledge of explicit form of perennials.
