A powerful application of symmetries is finding symmetry-invariant solutions of nonlinear differential equations. These solutions satisfy a reduced differential equation with one fewer independent variable. It is well known that a double reduction occurs whenever the starting nonlinear differential equation possesses a conservation law that is invariant with respect to the symmetry.
Recent work has developed a broad generalization of the double-reduction method by considering the space of invariant conservation laws with respect to a given symmetry. The generalization is able to reduce a nonlinear PDE in $n$ variables to an ODE with $m-n+2$ first integrals where $m$ is the dimension of the space of invariant conservation laws.
In this talk, a summary of the general multi-reduction method will be presented, with applications to obtaining invariant solutions of physically interesting PDEs.