Automorphisms of two-generator free groups and spaces of isometric actions on the hyperbolic plane

The automorphisms of a two-generator free group \mathsf F_2 acting on the space of orientation-preserving isometric actions of \mathsf F_2 on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action of a group \Gamma on \mathbb R ^3 by polynomial automorphisms preserving the cubic polynomial \kappa _\Phi (x,y,z) := -x^{2} -y^{2} + z^{2} + x y z -2 and an area form on the level surfaces \kappa _{\Phi}^{-1}(k)