On a complete set of commuting operators for ladder representations of the Lie algebra of $U(6,6)$

In connection with applications of tlie ladder representations the question about the completeness of a set of commuting operators with a definite physical interpretation is investigated. A suitable transformation of these operators is introduced, by means of which the problem under consideration is reduced to investigation of a system of functional equations. A number of theorems about this system are proved, from which the completeness of the mentioned set follows. As a result, we have established that the ladder representations of the Lie algebra of $U(6,6)$ can be given in terms of a physical complete set of commuting operators.