  Let $f, g: \mathbb{R}^2 \to \mathbb{R}$ be two submersion functions and
$\mathscr{F}(f)$ and $\mathscr{F}(g)$ be the regular foliations of
$\mathbb{R}^2$ whose leaves are the connected components of the levels sets of
$f$ and $g$, respectively. The topological equivalence of $f$ and $g$ implies
the topological equivalence of $\mathscr{F}(f)$ and $\mathscr{F}(g)$, but the
converse is not true, in general. In this paper, we introduce the class of
linear-like submersion functions, which is wide enough in order to contain
non-trivial behaviors, and provide conditions for the validity of the converse
implication for functions inside this class. Our results lead us to a complete
topological invariant for topological equivalence in a certain subclass of
linear-like submersion functions.
