First off, I know that there's no general algorithm for determining if there's a solution to a general Diophantine equation, much less a system.

However, I'm wondering if there is an algorithm for solving a Diophantine system of linear and quadratic equations? In fact, I have a system which is "sparse" in some sense (the linear equations are all the sum of three variables equals a number, and all the same number, and the quadratics aren't much worse).

If so, then can it be extended to the case of countably many variables?