Yes zero coordinates are allowed. However in my example matrix above there is no such vector I believe even with zero coordinates. The problem comes originally from a version of group testing I have become interested in.

I didn't properly express the requirements of my problem in this problem so have posted a follow up question. In particular, the elements of the vector all have to have the same absolute value.

I see what is going on now. (The answer to my question is (4,2,-2,-2)). Thank you.

That's very interesting, thank you. Just for my interest, which non-zero integer vector is in the null space of $ M = \begin{pmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 1\\ 0 & 1 & 0 & 1\ \end{pmatrix}.$ ?

I am not sure it is the same. I am asking if there is a non-zero vector with only integer coordinates in the null space. In my case all operations are over $\mathbb{Z}$ and I don't see the mapping to your examples.