We study an ordinary differential equation arising from the reduction of the classical sine‑Gordon equation to a self‑similar variable. By introducing a new dependent variable \ (f (z) \) and independent variable \ (z=- y\), the problem is reduced to a nonlinear second‑order ODE on the Riemann surface of \ (1-z²\). We rigorously investigate local analyticity, singular points (removable zeros and movable branch points), the behaviour at the fixed singular points \ (z=1\), and local analytic continuation through these points. Recurrence formulas for the coefficients of the power series at an arbitrary point are obtained. The role of the \ (z\) -representation as a tool for branch analysis is discussed, and it is pointed out that for global numerical construction of the solution an adaptive power‑series method in the variable \ (t= x\) is preferable.
Anton Kalmykov (Fri,) studied this question.