Solving multilinear integer equations

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The given equation can be rewritten as $$3(2y+1)(2z+1)=(2x+1)(2w+1),$$ which suggests that solutions can be generated by first fixing values of $y,z$ and thus $n:=3(2y+1)(2z+1)$, then running $d$ over the divisors of $n$, and setting $x=\frac{d-1}2$ and $w=\frac{n/d-1}2$ .

You can divide by $2$ and solve for $w$. So the set of all solutions is parametrized by $(x,y,z)\in\mathbb{Z}^3$ arbitrary.

This is NOT a 'real' answer - achrzesz had the real idea. Maybe he should post it for credit. This is just it in Sage.

sage: (x,y,z,w) = var('x,y,z,w')
sage: solve([12*(y*z) + 6*(y+x) +2 -4*x*y -2*(w+x) == 0,y==y], x,y,z,w)
[[x == r6, y == r7, z == r8, w == -(2*r6 - 3)*r7 + 6*r7*r8 + 2*r6 + 1]]

You may ask why we need the dummy equation y==y. See this Trac ticket.

This does not answer the question of how to treat this as a non-linear Diophantine equation. I seem to recall that this is undecidable, but maybe I'm misinterpreting your question. You can always just do something really naive like loop through all integers. I don't know that we have anything more efficient that that for integers.