A certain generalization of S. Lie's problem on translation surfaces

The problem of S. Lie on the translation surfaces is to find all the translation surfaces with two or more translation nets. It was solved by S. Lie, H. Poincaré and other mathematicians. A more general problem is to find all the surfaces admitting two or more conjugate nets of conic lines (the translation net consists of cylindric lines). A detailed survey of results obtained by various authors in this problem is given.