  We show that there exist 35 choices for the coordinate transformation each
leading to a potential for which the stationary Schr\"odinger equation is
exactly solvable in terms of the general Heun functions. Because of the
symmetry of the Heun equation with respect to the transposition of its
singularities only eleven of these potentials are independent. Four of these
independent potentials are always explicitly written in terms of elementary
functions, one potential is given through the Jacobi elliptic sn-function, and
the others are in general defined parametrically. Nine of the independent
potentials possess exactly or conditionally integrable hypergeometric
sub-potentials for which each of the fundamental solutions of the Schr\"odinger
equation is written through a single hypergeometric function. Many of the
potentials possess sub-potentials for which the general solution is written
through fundamental solutions each of which is a linear combination of two or
more Gauss hypergeometric functions. We present an example of such a potential
which is a conditionally integrable generalization of the third exactly
solvable Gauss hypergeometric potential.
