  The discussion is limited to first-class parametrized systems, where the
definition of time evolution and observables is not trivial, and to finite
dimensional systems in order that technicalities do not obscure the conceptual
framework. The existence of reasonable true, or physical, degrees of freedom is
rigorously defined and called {\em local reducibility}. A proof is given that
any locally reducible system admits a complete set of perennials. For locally
reducible systems, the most general construction of time evolution in the
Schroedinger and Heisenberg form that uses only geometry of the phase space is
described. The time shifts are not required to be 1symmetries. A relation
between perennials and observables of the Schroedinger or Heisenberg type
results: such observables can be identified with certain classes of perennials
and the structure of the classes depends on the time evolution. The time
evolution between two non-global transversal surfaces is studied. The problem
is posed and solved within the framework of the ordinary quantum mechanics. The
resulting non-unitarity is different from that known in the field theory
(Hawking effect): state norms need not be preserved so that the system can be
lost during the evolution of this kind.
