Holomorphic germs and the problem of smooth conjugacy in a punctured neighborhood of the origin

We consider germs of conformal mappings tangent to the identity at the origin in $\mathbf{C}$. We construct a germ of a homeomorphism which is a $C^{\infty}$ diffeomorphism except at the origin conjugating these holomorphic germs with the time-one map of the vector field $V(z)=z^{m}\tfrac{\partial}{\partial z}$. We then show that, in the case $m=2$, for a germ of a homeomorphism which is real-analytic in a punctured neighborhood of the origin, with real-analytic inverse, conjugating these germs with the time-one map of the vector field exists if and only if a germ of a biholomorphism exists.