  Sums of $M$ consecutive squared integers $\left(a+i\right)^{2}$ equaling
squared integers (for $a\geq1$, $0\leq i\leq M-1$) yield certain linear
groupings of pairs $\left(a_{1},a_{2}\right)$ of $a$ values for successive same
values of $M$ when these are linked by $a_{1}+a_{2}=\mu M+1$ with
$\mu=\left(\eta/\delta\right)\in\mathbb{\mathbb{Q}}^{+}$. In this paper,
congruent conditions on $M,\eta,\delta$, and on the difference
$\left(a_{2}-a_{1}\right)$ are demonstrated for these linear groupings to hold.
It is found that $\eta\equiv1\left(mod\,2\right)$ and $\delta\equiv0,1$ or
$5\left(mod\,6\right)$, and if $\delta\equiv0\left(mod\,6\right)$,
$M\equiv0\left(mod\,12\right)$, while if $\delta\equiv1$ or
$5\left(mod\,6\right)$, $M\equiv2$ or $11\left(mod\,12\right)$ with $a_{1}$ and
$a_{2}$ being of different or same parities.
