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