  Inspired by recent work of Carlson, Friedlander and Pevtsova concerning
modules for $p$-elementary abelian groups $E_r$ of rank $r$ over a field of
characteristic $p > 0$, we introduce the notions of modules with constant
$d$-radical rank and modules with constant $d$-socle rank for the generalized
Kronecker algebra $\mathcal{K}_r = k\Gamma_r$ with $r \geq 2$ arrows and $1
\leq d \leq r-1$. We study subcategories given by modules with the equal
$d$-radical property and the equal $d$-socle property. Utilizing the
Simplification method due to Ringel, we prove that these subcategories in
$\mathrm{mod} \ \mathcal{K}_r$ are of wild type. Then we use a natural functor
$\mathfrak{F} \colon \mathrm{mod} \ \mathcal{K}_r \to \mathrm{mod} \ kE_r$ to
transfer our results to $\mathrm{mod} \ kE_r$.
