local projective parametrization, with center at $P$, of the projective curve ${\cal C}^*$.

Thus, from $\psi_i(t):=\varphi_i(t^N),\,\,i=1,\ldots,N$ ($\varphi_i$ are the $N$ different series in the conjugacy class of $\varphi$), we obtain $N$ equivalent local projective parametrizations, $(1:\psi_i(t):t^N)$ (note that they are equivalent since $\varphi_1,\ldots\varphi_N$ belong to the same conjugacy class). Therefore, the leaves of $B$ are all associated to a unique infinity place.

Conversely, from a given infinity place defined by a local projective parametrization $(1:\psi(t):t^N)$ (see Theorem 2.5.3 in [@SWP]), we obtain $N$ Puiseux series, $\varphi_j(t)=\psi(c_j{t}^{1/N})$, $c_j^N=1$, that provide different expressions $r_j(z)=z\varphi_j(z^{-1}),\,j=1,\ldots,N$. Hence, the infinity branch $B$ is defined by the leaves $L_j=\{(z,r_j(z))\in
{\Bbb C}^2: \,z\in {\Bbb C},\,|z|>M\},\,\,j=1,\ldots,N.$

From the above discussion, we deduce that there exists a one-to-one relation between infinity places and infinity branches. In addition, we can say that each infinity branch is associated to a unique infinity point given by the center of the corresponding infinity place. Reciprocally, taking into account the above construction, we get that every infinity point has associated, at least, one infinity branch. Hence, every algebraic plane curve has, at least, one infinity branch. Furthermore, every algebraic plane curve has a finite number of branches.

Observe that the above construction can be 