  We find a new class of algebraic geometric solutions of Heun's equation with
the accessory parameter belonging to a hyperelliptic curve. Dependence of these
solutions from the accessory parameter as well as their relation to Heun's
polynomials is studied. Methods of calculating the algebraic genus of the
curve, and its branching points, are suggested. Monodromy group is considered.
Numerous examples are given.
