Integrable mappings of the plane preserving biquadratic invariant curves II

We review recently introduced curve-dependent McMillan maps which are mappings of the plane that preserve biquadratic foliations. We show that they are measure preserving and thus integrable. We discuss the geometry of these maps including their fixed points and their stability. We consider the normal forms of symmetric and asymmetric biquadratic curves and the normal forms for their associated McMillan maps. We further discuss the dynamics on biquadratic curves by considering the possibility of parametrizing them by Jacobian elliptic or rational functions.