Linear time-varying differential-algebraic  equations with symmetries are studied.
The structures that we address are self-adjoint and skew-adjoint systems.
Local and global canonical forms under congruence are presented and used
to classify the geometric properties of the flow associated with the differential
equation as symplectic or generalized orthogonal flow. As applications, the results
are applied to the analysis of dissipative Hamiltonian systems arising from circuit
simulation and incompressible flow.