In general, how do you solve the following kind of problems borrowing techniques from Group Theory?
Describe all points (if any) in the affine integral lattice $$ \mathcal{L} = \{(x, y, z, t) : x + y + z + t = 5 \text{ and } x - z \equiv 0 \mod 12\} \subset \mathbb{Z}^4$$
In this simple case, we can readily describe $\mathcal L$ as image of a $\Bbb Z^3$: We can pick $x,y\in \Bbb Z$ arbitrarily, then $z=x+12w$ with $w\in \Bbb Z$ and determine $t$ from $x+y+z+t=5$. So $$\mathcal L=\{\,(x,y,x+12w,5-2x-y-12w)\mid (x,y,w)\in\Bbb Z^3\,\}. $$