  Automorphic fundamental solutions and, more generally, solutions of
automorphic differential equations, play a key role in the
Diaconu-Garrett-Goldfeld prescription for spectral identities involving moments
of L-functions as well as other applications, including an explicit formula
relating the number of lattice points in a symmetric space to the automorphic
spectrum. In this paper we discuss two cases in which the automorphic
fundamental solution exhibits branching: pathwise meromorphic continuations may
differ by a term involving an Eisenstein series.
