Singular metrics and associated conformal groups underlying differential operators of mixed and degenerate types

For partial differential equations of mixed elliptic-hyperbolic and degenerate types which are the Euler-Lagrange equations for an associated Lagrangian, we examine an associated metric structure which becomes singular on the hypersurface where the operator degenerates. In particular, we show that the ``non-trivial part'' of the complete symmetry group for the differential operator (calculated in a previous paper by D. Lupo and K. R. Payne [Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types. Duke Math. J. (2005)]) corresponds to a group of local conformal transformations with respect to the metric away from the metric singularity and that the group extends smoothly across the singular surface. In this way, we define and calculate the conformal group for these operators as well as for lower order singular perturbations which are defined naturally by the singular metric.