In this paper we consider three problems concerning systems of vector exponentials. In the first part we prove a conjecture of V. Komornik raised in  on the independence of the movement of a rectangular membrane in different points. It was independently proved by M. Horváth  and S. A. Avdonin (personal communication). The analogous problem for the circular membrane was partly solved in  — the complete solution is given in . In the second part we fill in a gap in the theory of Blaschke-Potapov products developed in the paper  of Potapov. Namely we prove that the Blaschke-Potapov product is determined by its kernel sets up to a multiplicative constant matrix. In the third part of the present paper we give a multidimensional generalization of the notion of sine type function developed by Levin ,  and by our generalization we prove the multidimensional variant of the Levin-Golovin basis theorem , .