On the unitary part of isometries in commuting, completely non doubly commuting pairs

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.