  Let $Y=(f,g,h):\mathbb{R}^{3} \to \mathbb{R}^{3}$ be a $C^{2}$ map and let
$\spec(Y)$ denote the set of eigenvalues of the derivative $DY_p$, when $p$
varies in $\mathbb{R}^3$. We begin proving that if, for some $\epsilon>0,$
$\spec(Y)\cap (-\epsilon,\epsilon)=\emptyset,$ then the foliation
$\mathcal{F}(k),$ with $k\in \{f,g,h\},$ made up by the level surfaces
$\{k={\rm constant}\},$ consists just of planes. As a consequence, we prove a
bijectivity result related to the three-dimensional case of Jelonek's Jacobian
Conjecture for polynomial maps of $\mathbb{R}^n.$
