Solutions of Heun's general equation and elliptic Darboux equation

New solutions for the elliptic Darboux equation are obtained as particular cases of constructed Heun's general equation. We consider two groups power series expansions and new in Gauss hypergeometric functions. The convergence one group is determined by means ratio tests infinite series, while other designed to solve problems which admit finite-series solutions. Actually, we envisage periodic quasi-exactly solvable potentials stationary one-dimensional Schr\"odinger reduced In general, finite- infinite-series from However, show that admits additional terms functions a family associated Lam\'e used band theory solids. For each solution find well four bounded convergent all values independent variable. Finally it possible use transformations variables order generate out Heun