Let $k$ be any number field, and suppose we want to study the $k$-rational points on $$y^2 x = f(x),$$ where $f$ is a polynomial of degree greater or equal than 3. In other words, $y^2 x = f(x)$ is a sort of twisted elliptic or hyperelliptic curve. Question: What are the techniques available to tackle equations like $y^2 x = f(x)$?