The H-G Theorem say that if the singularity is hyperbolic (all the eigenvalues of the "linearized" system have a not null real part), then in an environment of the singularity the phase portrait is topologically conjugated to that of the linearized system.
In all the books I have read, one of the hypotheses assumed in the proof of this theorem is that the singular point is at the origin. However, intuition tells me that the Theorem also works with singularities that are not at the origin. Is this so?
Yes, certainly, since you can just apply the simple change of variables $y=x-x^*$ to move the singular point from $x=x^*$ to $y=0$.