Pell surfaces

In 1826 Abel started the study of the polynomial Pell equationx² − g(u)y² = 1. Its solvability in polynomials x(u), y(u) depends on a certain torsionpoint on the Jacobian of the hyperelliptic curve v² = g(u). In this paper westudy the affine surfaces defined by the Pell equations in 3-space with coordinatesx, y, u, and aim to describe all affine lines on it. These are polynomial solutions ofthe equation x(t)² − g(u(t))y(t)² = 1. Our results are rather complete when thedegree of g is even but the odd degree cases are left completely open. For evendegrees we also describe all curves on these Pell surfaces that have only 1 placeat infinity.