<p>This work investigates a Bitsadze-Samarskii type problem for the Viscous-Transonic equation. The uniqueness of the solution is proven using the energy integral method. It is shown that when the uniqueness conditions do not hold, nontrivial solutions to the homogeneous problem exist. The existence of the solution is established via the method of separation of variables. The solution is presented as an infinite series, with justification for term-by-term differentiation with respect to all variables. In proving the uniform convergence of the series, we show that the “small denominator” is not zero.</p>