Setup If we are given $a,b,c,d$, then the system of equations $$ \begin{align} p+r & =a, \tag {1a}\\ pr+qs & =b, \tag {1b}\\ p(s+1)+qr+r & =c, \tag {1c}\\ 2pr+qs+2 & =d \tag {1d}\\ \end{align} $$
can be solved for $p,q,r,s$, not necessarily integers, because the equations are all independent. The outline is we find $pr = d - b - 2$ by solving Eqns. $(1b), (1d)$ then use it with $(1a)$ to solve for $p,r$. A similar approach is used for solving for $q,s$.
The linear diophantine equation
$$ -2a+2b-2c+d = n \tag 2 $$
has an infinite number of solutions in integers for $a,b,c,d$.
Question: Given only $n$, find all integer solutions for $p,q,r,s,a,b,c,d$ simultaneously satisfying Eqns. $(1)$ and $(2)$. How do we solve this?
Approach tried: I was able to find a particular solution
$$ \begin{align} b & =d - 2 , \\ p & =0, \\ q & = {{c - a} \over {a}}, \\ r & =a, \\ s & =-{{a (d - 2)} \over {a - c}} = {{b} \over {q}}, \\ a(a - c) & ≠0. \end{align} $$