To Fuchsian partial differential equations in the sense of M. S. Baouendi and C. Goulaouic, which is a natural extension of ordinary differential equations with regular singularity at a point, all the solutions in a complex domain are constructed along the same line as the method of Frobenius to ordinary differential equations, without any assumptions on the characteristic exponents. The same idea can be applied to Fuchsian hyperbolic equations considered by H. Tahara.
"The method of Frobenius to Fuchsian partial differential equations." J. Math. Soc. Japan 52 (3) 645 - 672, July, 2000. https://doi.org/10.2969/jmsj/05230645