Integrable mappings of the plane preserving biquadratic invariant curves

We provide a general framework to construct integrable mappings of the plane that preserve a one-parameter family B(x, y, K) of biquadratic invariant curves where parametrization by K is very general. These mappings are reversible by construction (i.e. they are the composition of two involutions) and can be shown to be measure preserving. They generalize integrable maps previously given by McMillan and Quispel, Roberts and Thompson. By considering a transformation of the case of the symmetric biquadratic to a canonical form, we provide a normal form for the symmetric integrable map acting on each invariant curve. We give a Lax pair for a large subclass of our symmetric integrable maps, including at least a 10-parameter subfamily of the 12-parameter symmetric Quispel-Roberts-Thompson maps.