  We consider the modified Monge-Kantorovich problem with additional
restriction: admissible transport plans must vanish on some fixed functional
subspace. Different choice of the subspace leads to different additional
properties optimal plans need to satisfy. Our main results are quite general
and include several important examples. In particular, they include
Monge-Kantorovich problems in the classes of invariant measures and
martingales. We formulate and prove a criterion for existence of a solution, a
duality statement of the Kantorovich type, and a necessary geometric condition
on a support of optimal measure, which is analogues to the usual
$c$-monotonicity.
