An existence result for the Shabat equation (Q1849455)

The author deals with the Shabat equation \[ f'(z)+q^2f' (qz)+ f^2(z)- q^2f^2(qz)= \mu, \] which is useful for the study of the \(q\)-deformed Heisenberg-Weyl algebra. The existence of analytic solutions in the critical case, where the complex parameter \(q\) belongs to the unit circle, is investigated. It is shown that, for any \(q\) belonging to a certain subset of the unit circle with measure \(2\pi\), the Shabat equation has a unique analytic solution in a neighbourhood of the origin with any prescribed value at the origin.