  We consider a two-dimensional commutative algebra B over the field of complex
numbers. The algebra B is associated with the biharmonic equation. For
monogenic functions with values in B, we consider a Schwartz-type boundary
value problem (associated with the main biharmonic problem) for a half-plane
and for a disk of the biharmonic plane. We obtain solutions in explicit forms
by means of Schwartz-type integrals and prove that the mentioned problem is
solvable unconditionally for a half-plane but it is solvable for a disk if and
only if a certain natural condition is satisfied.
