Particular solutions of generalized Euler-Poisson-Darboux equation

In this article we consider the generalized Euler-Poisson-Darboux equation $${u}_{tt}+\frac{2\gamma }{t}{{u}_{t}}={u}_{xx}+{u}_{yy} +\frac{2\alpha }{x}{{u}_{x}}+\frac{2\beta }{y}{{u}_y},\quad x>0,\;y>0,\;t>0.$$ We construct particular solutions in an explicit form expressed by the Lauricella hypergeometric function of three variables. Properties of each constructed solutions have been investigated in sections of surfaces of the characteristic cone. Precisely, we prove that found solutions have singularity $1/r$ at $r\to 0$, where ${{r}^2}={{( x-{{x}_0})}^2}+{{( y-{{y}_0})}^2}-{{( t-{{t}_0})}^2}$