  In [{\it American J. Mathematics}, 124(2002), 107--127] we proved that for a
germ of $C^\infty$ hyperbolic diffeomorphisms $F(x)=Ax+f(x)$ in $(\mathbb
R^n,0)$, if $A$ has a real logarithm with its eigenvalues weakly nonresonant,
then $F(x)$ can be embedded in a $C^\infty$ autonomous differential system. Its
proof was very complicated, which involved the existence of embedding periodic
vector field of $F(x)$ and the extension of the Floquet's theory to nonlinear
$C^\infty$ periodic differential systems. In this paper we shall provide a
simple and direct proof to this last result.
  Next we shall show that the weakly nonresonant condition in the last result
on the real logarithm of $A$ is necessary for some $C^\infty$ diffeomorphisms
$F(x)=Ax+f(x)$ to have $C^\infty$ embedding flows.
  Finally we shall prove that a germ of $C^\infty$ hyperbolic diffeomorphisms
$F(x)=Ax+f(x)$ with $f(x)=O(|x|^2)$ in $(\mathbb R^2,0)$ has a $C^\infty$
embedding flow if and only if either $A$ has no negative eigenvalues or $A$ has
two equal negative eigenvalues and it can be diagonalizable.
