  Spacelike surfaces with the same mean curvature in $\mathbb{R}^3$ and
$\mathbb{L}^3$ are locally described as the graph of the solutions to the
$H_R=H_L$ surface equation, which is an elliptic partial differential equation
except at the points at which the gradient vanishes, because the equation
degenerates. In this paper we study precisely the critical points of the
solutions to such equation. Specifically, we give a necessary geometrical
condition for a point to be critical, we obtain a new uniqueness result for the
Dirichlet problem related to the $H_R=H_L$ surface equation and we get a
Heinz-type bound for the inradius of the domain of any solution to such
equation, improving a previous result by the authors. Finally, we also get a
bound for the inradius of the domain of any function of class $\mathcal{C}^2$
in terms of the curvature of its level curves.
