  In this paper we present results concerning the stabilizer $G_f$ in
$\mathrm{GL}(2,q^n)$ of the subspace $U_f=\{(x,f(x))\colon x\in\mathbb
F_{q^n}[x]\}$, $f(x)$ a scattered linearized polynomial in $\mathbb
F_{q^n}[x]$. Each $G_f$ contains the $q-1$ maps $(x,y)\mapsto(ax,ay)$,
$a\in\mathbb F_{q}^*$. By virtue of the results of Beard (1972) and Willett
(1973), the matrices in $G_f$ are simultaneously diagonalizable. This has
several consequences: $(i)$ the polynomials such that $|G_f|>q-1$ have a
standard form of type $\sum_{j=0}^{n/t-1}a_jx^{q^{s+jt}}$ for some $s$ and $t$
such that $(s,t)=1$, $t>1$ a divisor of $n$; $(ii)$ this standard form is
essentially unique; $(iii)$ for $n>2$ and $q>3$, the translation plane $\cal
A_f$ associated with $f(x)$ admits nontrivial affine homologies if and only if
$|G_f|>q-1$, and in that case those with axis through the origin form two
groups of cardinality $(q^t-1)/(q-1)$ that exchange axes and coaxes; $(iv)$ no
plane of type $\cal A_f$, $f(x)$ a scattered polynomial not of pseudoregulus
type, is a generalized Andr\'e plane.
