  The problem of existence of standard (i.e. product-type) invariant MASAs for
endomorphisms of the Cuntz algebra O_n is studied. In particular endomorphisms
which preserve the canonical diagonal MASA D_n are investigated. Conditions on
a unitary in O_n equivalent to the fact that the corresponding endomorphism
preserves D_n are found, and it is shown that they may be satisfied by
unitaries which do not normalize D_n. Unitaries giving rise to endomorphisms
which leave all standard MASAs invariant and have identical actions on them are
characterized. Finally some properties of examples of finite-index
endomorphisms of O_n given by Izumi and related to sector theory are discussed
and it is shown that they lead to an endomorphism of O_2 associated to a matrix
unitary which does not preserve any standard MASA.
