  There are considered isometries on a Hilbert space. By the Wold theorem any
isometry can be decomposed into a unitary operator and a unilateral shift. For
a pair of isometries, even commuting, a maximal subspace reducing one isometry
to a unitary operator might not reduce the other isometry. In the paper are
considered pairs of commuting isometries which are completely non doubly
commuting. For such pairs there are no nontrivial subspaces reducing both
isometries and one of them to a unitary operator. The results describe a
unitary part of an isometry in such a pair.
