Solve $a^2+b^2+c^2+d^2+2ab+2bc+2cd-2ca-2ad-2db=N^2$ I am looking for a characterization of the solutions of 
$$a^2+b^2+c^2+d^2+2ab+2bc+2cd-2ca-2ad-2db=N^2$$ in positive integers.

I suppose that you mean
$$
a^2 + b^2+c^2+d^2 + 2ab - 2ac - 2ad - 2bc - 2bd +  2cd =N^2.
$$
Since the left hand side is $(a+b-c-d)^2$, we are done.
