Special Lagrangian submanifolds of Calabi-Yau manifolds have become important for applications to string theory and mirror symmetry, but they remain poorly understood, particularly in as regards their singular loci and global behavior. Constructing explicit examples has been an important project in developing the theory and the standard approach has been to apply some sort of symmetry reduction. Since the group actions used in the reduction process are usually not free, it becomes important to understand what happens near singular orbits, in particular, to understand the behavior of group invariant special Lagrangian submanifolds that meet the fixed locus of the action. This frequently leads to interesting singular PDE problems and this talk will describe a class of such examples constructible by invoking an existence theorem for such singular PDE.