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