A linear Fuchsian equation with variable indices

We construct three types of solutions for a Fuchsian equation withvariable indices: (1) branched solutions involving logarithms ofthe time variable $t$; (2) solutions involving $t^x$, where $x$ isa space variable; and, (3) for a model case, exact solutionsinvolving hypergeometric functions. These three solutions havecompletely different singularities. The constructions are given ina form suitable for application to more general equations. As anillustration, we resolve in particular an apparent discrepancybetween two recent results on this problem