# Detecting blocks in non-linear system of equations

When solving systems of non-linear equations using Newton's method, it is often observed that the system has an independent sub-system, e.g. :

$$f(x,y) = 0$$ $$g(x,y) = 0$$ $$h(x,y,z) = 0$$

If I am not mistaken, you can (and should) solve first for $$x$$ and $$y$$ and then subsequently solve for $$z$$. Now when the dimension of the system increases, it is not so easy to spot such independent sub-systems. Are there algorithms adapted to this issue ?

In my specific case, the Jacobian of the system has a fixed structure so even a slow algorithm for detection would be largely beneficial when solving the non-linear system using Newton's method. I feel that this is a common topic in solving systems of equations, so I am open to be redirected to additional ressources that are relevant.