Singular

I'm completely new to Singular, and wish to obtain "non-trivial" solutions to a system of equations. More precisely, I create a ring

and an ideal

and its Gröbner basis. I want to find u(...),v(...),x(...),y(...) that satisfy the equations in i. Two problems:
(1) i has positive dimension (actually projective of dimension 11, degree 25), so does not work
(2) there are some trivial solutions, for example u=x,v=y which I want to avoid

I tried to specialize some variables at random (by defining ideal j = i,u(1)-1,...), but didn't manage to find a non-trivial 0-dimensional ideal in this manner, so something more systematic is welcome!

Avoiding trivialities can be done by using saturation, so relative to ideals such as ideal j=u-v. This can be found in Singular.

Finding varieties that are not 0-dimensional from a lex GB is more problematic.
These are usually not implemented.

Look up elimination and extension on the web.
I wrote Macaulay2 code for this in a 2019 paper (on arXiv) using multi-homogeneous coordinates rather than projective ones,
as I thought this did what was in Cox, Little, O'Shea without having to deal with exceptions.
But I'm sure there are other papers on the subject as well.