Linearized transverse equation for a surface thermostat ($\dot{y}=z,\dot{z}=-(K-H\lambda +{\lambda }^2)y+(V\lambda )z$)

Write a tangent variation of a surface thermostat as $aF+yH+zV$ in the canonical frame. The commutators $[F,H]=-\lambda F+(K-H\lambda +{\lambda }^2)V$ and $[F,V]=-H-(V\lambda )V$ give $\dot{a}=\lambda y$ and the stated transverse equation. Near the vertical projective direction, $r=y/z$ obeys $\dot{r}=1-(V\lambda )r+(K-H\lambda +{\lambda }^2)r^2$.