Horizontal gradient of polynomial functions for the standard Engel structure on mathbb R⁴

We investigate the set $V_f$ of horizontal critical points of a polynomial function $f$ for the standard Engel structure defined by the 1-forms $\omega_3=dx_3-x_1dx_2,$ $\omega_4=dx_4-x_3dx_2$, endowed with the sub-Riemannian metric $g_{SR}=dx_1^2+dx_2^2$. For a generic polynomial, we show that the intersection of any fiber of $f$ and $V_f$ does not contain a horizontal curve. Then we prove that each trajectory of the horizontal gradient of $f$ approaching the set $V_f$ has a limit.