Dynamical systems whose symplectic structure degenerates, becoming
noninvertible at some points along the orbits are analyzed. It is shown that
for systems with a finite number of degrees of freedom, like in classical
mechanics, the degeneracy occurs on domain walls that divide phase space into
nonoverlapping regions each one describing a nondegenerate system, causally
disconnected from each other. These surfaces are characterized by the sign of
the Liouville's flux density on them, behaving as sources or sinks of orbits.
In this latter case, once the system reaches the domain wall, it acquires a new
gauge invariance and one degree of freedom is dynamically frozen, while the
remaining degrees of freedom evolve regularly thereafter.