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I'm trying to understand the inhomogeneous SIS problem and I'm came across to a scenario that I don't know how to evaluate.

Let $A,B \in \mathbb{Z}_q^{n\times m}$ be two random matrixes and $u,v \in \mathbb{Z}_q^m$ be two vectors of small norm $||u||,||v||<\sigma$, such that $A.u=B.z$.

How easy would be to find another pair $w,y \in \mathbb{Z}_q^m$, of short vectores that satisfy $A.w=B.y$, assuming that both $A.w = 0$ and $B.y=0$ are hard SIS problems?

I've tried to find this problem online but I couldn't find anything about it. I'm not sure if it is because this not a hard problem or if it is because I don't know the name by which it is known.

Can't somebody give me some pointers to follow?

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Answering my own question.

I guess the problem has no specific name because it is not different from the SIS problem.

Let $C=[A|-B]$ and $s=[u|v]$ then the problem $A.u=B.v$ is equivalent to $C.s=0$

Sorry for the trouble.

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