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 equationf′(z)+q2f′(qz)+f2(z)−q2f2(qz)=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|q|=1 has a non-trivial analytic solution f(z)f(z) under the Brjuno condition.