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 \(\kappa _{\Phi}^{-1}(k)\).