Branch structure and local analytic continuation of the solution of the Amsler equation in the chart \(z=-cos y\) - Construction of a combined fast solver

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 \(z=-cos y\), the problem is reduced to a nonlinear second‑order ODE on the Riemann surface of \(√1-z^2\). 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 \(t=ln x\) is preferable.