Two things:

1. Your Cartesian equation can be explicitly solved for $y$, so you can either a. use Plot3D[] and rotate the surface afterwards; or b. use ParametricPlot3D[] directly.

2. You can directly construct an appropriate ColorFunction by using Darker[] with some function that asymptotically approaches $1$:

Thus:

ParametricPlot3D[{x, z^2 + 1/x^3, z}, {x, -10, 10}, {z, -10, 10}, 
                 ColorFunction -> Function[{x, y, z},
                                           If[y == 0, Green,
                                              Darker[If[y > 0, Blue, Red],
                                                     Tanh[Abs[y]/10]]]], 
                 ColorFunctionScaling -> False, Mesh -> False, PlotPoints -> 45, 
                 PlotRange -> 10]

(I'm not able to run this on Mathematica at the moment, so please adjust the ColorFunction and PlotPoints settings to your taste.)

Hi Hans, thank you very much for your answear. I haven't been using mathematica for a while, and I was and I am a complete beginner with this programming language.

Back to my question: I knew there was a singularity in x == 0, but the plot was intented to rapresent a thing called Phase Portrait. Within this context, I have to show three zones for the surface, one of whom where the transition happens at y=0. This should be achieved by using your p3 proposal and setting up different conditions. Since I need the top view, which option should I be using? I used ViewPoint but it gives an unpleasant 3D pespective, like in ControurPlot.

Once again, thank you very much for your help and the time you dedicated me :)

First: x == 0 is a singularity, your surface is not defined.

Second: look at this, does that help (I avoided x == 0 , therefore two plots)?

p1 = ContourPlot3D[
  z^2 - y + 1/x^3 == 0, {x, -10, -.2}, {y, -10, 10}, {z, -10, 10}, 
  Mesh -> False,
  ColorFunction -> Function[{x, y, z}, If[y < 0, Red, Blue]],
  ColorFunctionScaling -> False]
p2 = ContourPlot3D[
  z^2 - y + 1/x^3 == 0, {x, .2, 10}, {y, -10, 10}, {z, -10, 10}, 
  Mesh -> False,
  ColorFunction -> Function[{x, y, z}, If[x > 0, Magenta, Blue]],
  ColorFunctionScaling -> False]
Show[p2, p1, PlotRange -> All]