  Hartman-Grobman theorem states that there is a homeomorphism H sending the
solutions of the nonlinear system onto those of its linearization under
suitable assumptions. Many mathematicians have made contributions to prove
H\"older continuity of the homeomorphisms. However, is it possible to improve
the H\"older continuity to Lipschitzian continuity?
  This paper gives a positive answer. We formulate the first result that the
homeomorphism is Lipschitzian, but not $C^1$, while its inverse is merely
H\"{o}lder continuous, but not Lipschitzian. It is interesting that the
regularity of the homeomorphism is different from its inverse. Moreover, some
illustrative examples are presented to show the effectiveness of our results.
Further, motivated by our example, we also propose a conjecture, saying, the
regularity of the homeomorphisms is sharp and it could not be improved any
more.
