  We study stable subspaces of positive extremal maps of finite dimensional
matrix algebras that preserve trace and matrix identity (so-called bistochastic
maps). We have established the existence of the isometric-sweeping
decomposition for such maps. As the main result of the paper, we have shown
that all extremal bistochastic maps acting on the algebra of matrices of size
3x3 fall into one of the three possible categories, depending on the form of
the stable subspace of the isometric-sweeping decomposition. Our example of an
extremal atomic positive map seems to be the first one that handles the case of
that subspace being non-trivial. Lastly, we compute the entanglement witness
associated with the extremal map and specify a large family of entangled states
detected by it.
