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I learned that in general, solving systems of quadratic Diophantine equations is a difficult problem. But I wonder if there are special (and non-trivial) types of systems that are easier to handle.

For instance, I am interested in the following scenario:

  • There are $n$ unknowns and $(n-1)$ homogeneous quadratic Diophantine equations in the system. So each equation can be written as a quadratic form $x^TAx=0$ for a symmetric matrix $A$.
  • The matrix $A$ is zero-diagonal (and thus indefinite). The non-zero entries of $A$ are only $1$ or $-1$.
  • Each $A$ has a small rank.

I am not sure if the above situation is too specific. My main question is whether there are cases when the problem of solving a system of homogeneous quadratic Diophantine equations is easier. And I will appreciate any suggestion for references related to the questions.

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