I don't think it's fair to say that the Gröbner basis has nothing to say about holonomicity. For example, if the cotangent space $T^*X$ has dimension 4, and the Gröbner basis has dimension three then wouldn't it follow that $in_{(0,1)}(I)$ has three generators as well? In the generic case it seems very unlikey that this system is holonomic unless a miracle occurs, and the zeros in the third equation are dependent on the other two.

In Sato's book on Gröbner bases he seems to define holonomicity by the requirement that the initial ideal in$_{(0,1)}(I)$ has dimension at most $n$. But doesn't the Gröbner basis span in$_{(0,1)}(I)$? (Edit: Ah, I see, it spans $gr(I)$ while in$_{(0,1)}(I)\subset gr(I)$ are the highest weights). The reason I would like to say this system is not holonomic is because I was under the impression that a holonomic system is also involutive... isn't that what Kashiwara has shown?

I just came here to point out that, indeed, the ideal $I=D_X(\partial+x)+D_X(\partial+\partial')$ does not define a holonomic D-module because the Gr\"obner basis is not spanned by only two elements. The obstruction is that the pair does not form an S-pair (as in the reference you pointed to), which is akin to saying that the system is not involutive. I saw your edit before I could type it, so I'm awarding you the points for the most thorough answer, though I'm grateful to everyone who commented.

Thanks for the suggestions. I just discovered Gröbner's book while looking into it, so I'll check that one out first. I'm still stumped on this problem, though, since it would seem that even your advice amounts to looking at the principle symbols, yet, supposedly there is no nontrivial solution unless one drops the $z$ term, which is something that cannot be inferred from the principle symbol. Am I wrong that this system is holonomic? And if it's holonomic, shouldn't it have a nonzero solution?

Thank you for this insight! This is very interesting and I'd like to read more if you can provide a reference. In this case, is it true that $r\times s=1\times 2$, so that the Characteristic ideal is simply $(p,\ p+p')=(0,0)$, which is itself isomorphic the Characteristic variety, $Ch(M)\subset T^*X$ for the module $M=D_X/I$, for $I=D_X(\partial+\beta(x))+D_X(\partial+\partial')$...?

Would you mind elaborating on this? How would it look in the current problem? I am a physicist, so I'm afraid I may need it spelled out a bit more. Specifically, I don't understand what the Ann function represents.

Thank you for the comment. That is indeed the correct way to go about it in the Frobenius picture when the equations are linear. However, in the second link the general case is considered micro locally. It seems that the second system does have a solution if only in the micro local sense of the term. I'm wondering if it implies there exists a hypersurface M⊂C2 on which a solution exists for restricted values (z,z′)∈M?