  Let $k$ be an algebraically closed field, $\mathop{char}(k) = p \geq 2$ and
$E_r$ be a $p$-elementary abelian group of rank $r \geq 2$. Let $(c,d) \in
\mathbb{N}^2$. We show that there exists an indecomposable module of constant
Jordan type $[1]^c [2]^d$ and Loewy length $2$ if and only if
$q_{\Gamma_r}(d,d+c) \leq 1$ and $c \geq r-1$, where $q_{\Gamma_r}(x,y) := x^2
+ y^2-rxy$ denotes the Tits form of the generalized Kronecker quiver
$\Gamma_r$. Since $p > 2$ and constant Jordan type $[1]^c [2]^d$ imply Loewy
length $\leq 2$, we get in this case the full classification of Jordan types
$[1]^c [2]^d$ that arise from indecomposable modules.
