A linear system of two equations with a quasiperiodic differentiation operator were considered in the
study. The study of the problem of the existence and uniqueness of solutions, periodic in one variable and quasiperiodic
in another variable with the same frequency basis is developed based on methods of the theory of multi-periodic
solutions of a system with multi-periodic input data, that can be real-analytically continued in each variable to some
complex neighborhood of the real axis. The sufficient conditions for the existence of a unique solution possessing the
properties of real-analyticity and quasiperiodicity with respect to variables and an upper estimate for a norm depending
on the width of the neighborhood of the real axis are established. A new research method that is applicable to solve this
problem, both in non-critical and critical cases is proposed.