We present a unified geometric framework that continuously deforms the symmetric Amsler surface into the pseudosphere (kink solution). The construction uses a five‑dimensional extension of the Time‑Shared Object (TSO) that incorporates a branch parameter selecting the solution of the reduced ODE. The GAE family with the s condition provides a path from the Amsler surface to the constant 2 degenerate flat metric, while a one‑parameter family of pendulum solutions connects the same constant to the kink. By smoothly joining these segments we obtain a C^ path in the extended space, demonstrating that the two classical pseudospherical surfaces are continuously deformable into one another. The paper details the limiting behaviour at s=0, 1, 2, 3, the construction of the 5D TSO, the degenerate metric connections, the piecewise and smooth paths, and concludes with a geometric picture of the unified parameter space.
Anton Kalmykov (2026) studied this question.