  We investigate the moduli spaces of one- and two-dimensional sheaves on
projective K3 and abelian surfaces that are semistable with respect to a
nongeneral ample divisor with regard to the symplectic resolvability. We can
exclude the existence of new examples of projective irreducible symplectic
manifolds lying birationally over components of the moduli spaces of
one-dimensional semistable sheaves on K3 surfaces, and over components of many
of the moduli spaces of two-dimensional sheaves on K3 surfaces, in particular,
of those for rank two sheaves.
