Let $G(z)$ and $S(z)=\sum_{k\geq 1}k s_k z^k$ as polynomial in $z$. Let $$X(z):=\frac{z}{ G(S(z))},\quad Y(z):=\frac{S(z)}{z} G(S(z)) \tag{$***$} $$ be a rational parametrization of a plane algebraic curve.
An example would be $$G(z)= 1+z, \quad S(z)=z^r $$ where $r$ is positive integer. I am mostly interested in the case when $d(X(z)):= \frac{\partial X(z)}{\partial z}$ have simple zeros that imply in my example that would imply $r\leq 3$.
What is the classification of plane curves that are having rational parametrization that is $X(z)$ and $Y(z)$ are a rational function of parameter $z$ but is not of the form (***)? Also I would restrict $d(X(z))$ to have simple zeros.
An immediate example of curves not of the from $(***)$ would be
$$X(z)= \frac{z^2+1}{z}, \quad Y(z)=\frac{z}{z^2-1} $$
Is there any geometric information about such a class of curves?