  We describe a way of solving a partial differential equation using the
differential invariants of its point symmetries. By first solving its quotient
PDE, which is given by the differential syzygies in the algebra of differential
invariants, we obtain new differential constraints which are compatible with
the PDE under consideration. Adding these constraints to our system makes it
overdetermined, and thus easier to solve. We focus on second-order scalar PDEs
whose quotients are first-order scalar PDEs. This situation occurs only when
the Lie algebra of symmetries of the second-order PDE is infinite-dimensional.
We apply this idea to several different PDEs, one of which is the Hunter-Saxton
equation.
