  Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a real homogeneous polynomial and
$S(f)$ be the group of diffeomorphisms $h:\mathbb{R}^2 \to \mathbb{R}^2$
preserving $f$, i.e. $f \circ h = f$. Denote by $S(f,r)$, $(0\leq r \leq
\infty)$, the identity path component of $S(f)$ with respect to the weak
Whitney $C^{r}_{W}$-topology. We prove that $S(f,\infty) = \cdots = S(f,1)$ for
all such $f$ and that $S(f,1) \not= S(f,0)$ if and only if $f$ is a product of
at least two distinct irreducible over $\mathbb{R}$ quadratic forms.
