Conservation laws that depend on functions and PDE reduction: Extending Noether $1\tfrac {1}{2}$

Abstract This paper develops methods for simplifying systems of partial differential equations (PDEs) that have families of conservation laws which depend on arbitrary functions of the independent or dependent variables. Cases are identified in which such methods can be combined with reduction using families of symmetries to give a multiple reduction; this is analogous to the double reduction of order for ordinary differential equations (ODE) with variational symmetries. Applications are given, including a widely used class of pseudoparabolic equations and several mean curvature equations.