C}^*$ is defined by $F(x:y:z)=$ $$y^5-4y^4x+4y^3x^2+2y^2z^2x-zy^2x^2+2z^2yx^2+2yzx^3+z^4x+z^3x^2\in {\Bbb R}[x,y,z].$$ Note that $P=(1:0:0)$ is an infinity point of ${\cal C}^*$. Let us compute the infinity branches associated to $P$. For this purpose, we consider the curve defined implicitly by the polynomial $g(y,z)=F(1:y:z)$, and we observe that $g(p)=0,$ where $p=(0,0)$.

We compute the series expansion for the solutions of $g(y,z)=0$. For this purpose, we use for instance the [algcurves]{} package included in the computer algebra system [Maple]{}. We get that: $$\varphi_1(z)=-1/2z^2+1/8z^4-1/8z^5+1/16z^6+1/16z^7+\cdots\in{\Bbb C}\ll z\gg,\quad \mbox{and}$$ $$\varphi_2(z)=\frac{-(-2z)^{1/2}}{2}-\frac{z}{8}  +\frac{27}{256} (-2 z)^{3/2}-\frac{7}{32} z^2+\frac{4057}{65536} (-2 z)^{5/2}+\cdots\in{\Bbb C}\ll z\gg.$$

That is, $g(\varphi_j(z), z)=0,\,j =1,2$ (see e.g. Section 2.5 in [@SWP]). Note that $\nu(\varphi_1)=1$, which implies that we only have one Puiseux series in the conjugacy class of $\varphi_1$. However, $\nu(\varphi_2)=2$ and then, we have the following conjugate Puiseux series in the conjugacy class of $\varphi_2$: $$\varphi_{2,1}(z)=\frac{-(-2z)^{1/2}}{2}-\frac{z}{8}  +\frac{27}{256} (-2 z)^{3/2}-\frac{7}{32} z^2+\frac{4057}{65536} (-2
z)^{5/2}+\cdots$$ $$\varphi_{2,2}(z)=\frac{+(-2z)^{1/2}}{2}-\frac{z}{8}  -\frac{27}{256} (-2 z)^{3/2}-\frac{7}{32} z^2-\frac{4057}{65536} (-2
z)^{5/2}+\cdots$$ Thus, we obtain two infinity branches: $$B_1=L_1=\{(z,r_1(z))\in {\Bbb C}^2: \,z\in {\Bbb C},\,|z|>M\},\qquad \mbox{where}$$ $$r_1(z)=z\varphi_1(z^{-1})=-1/(2z)+1/(8z^3)-1/(8z^4)+1/(16z^5)+1/(16z^6)+\cdots$$ and $B_2=L_{2,1}\cup L_{2,2}$, where $L_{2,i}=\{(z,r_{2,i}(z))\in {\Bbb C}^2:
\,z\in {\Bbb C},\,|z|>M\}$, $i=1,2$ and $$r_{2,1}(z)=z\varphi_{2,1}(z^{-1})=-\frac{(-2z)^{1/2}}{2}-\frac{1}{8}+\frac{ 27(-2z)^{-1/2}}{64}-\frac{7z^{-1}}{32}+\frac{ 4057(-2z)^{-3/2}}{4096}+\cdots.$$ $$r_{2,2}(z)=z\varphi_{2,2}(z^{-1})=+\frac{(-2z)^{1/2}}{2}-\frac{1}{8}-\frac{ 27(-2z)^{-1/2}}{64}-\frac{7z^{-1}}{32}-\frac{
4057(-2z)^{-3/2}}{4096}+\cdots.$$

In Figure \[F-ramas infinitas2\], we plot the curve ${\cal C}$ and some points of the infinity branches $B_1$ 