Small divisors problem in dynamical systems and analytic solutions of the Shabat equation

This paper is concerned with the existence of exact analytic solutions to the Shabat equation f ′ ( z ) + q 2 f ′ ( q z ) + f 2 ( z ) − q 2 f 2 ( q z ) = μ , z ∈ C , where μ and q are complex parameters. This equation is the simplest self-similar reduction of so-called dressing chain for constructing and analyzing exactly solvable Schrödinger equation. It is so relevant to the study of the q-oscillator algebra in quantum mechanics. The main aim of this paper is to investigate whether the Shabat equation with | q | = 1 has a non-trivial analytic solution f ( z ) under the Brjuno condition.