Given: a bivariate polynomial $P(u,v)$ such that $\{ (u,v): P(u,v)=0 \}$ is a (smooth) algebraic curve. Then $P(y,y')$ is autonomous first-order ODE for which local solutions always exist because it is always (at least locally) separable (provided that $P(u,v)=0$ is a smooth algebraic curve).
Question: What is the name of all special functions $y=f(x)$ such that $P(y,y')=0$ for some smooth algebraic curve $P(u,v)=0$?
For smooth algebraic curves of degree $2$, if $P$ is an ellipse, then $y = \sin(at)$ or $y\cos(bt)$. If $P$ is a hyperbola, then $y = \sinh(at)$ or $y = \cosh(bt)$. And if $P$ is a smooth cubic curve (i.e. degree 3), then the solution $y$ is a Weierstrass $\wp$ function. What is the name for general $y$?
In particular, a (smooth) algebraic curve of genus $n$ corresponds (at least in $\mathbb{CP}^2$) to an orientable compact surface of genus $n$, and thus each $y$, given that $P(u,v)=0$ is a smooth algebraic curve of genus $n$, can be defined as a function on an orientable compact surface of genus $n$. The fact that surfaces of genus $0$ and genus $1$ have abelian fundamental groups explains why (I think) the trigonometric functions and the Weierstrass $\wp$ functions are periodic; since the fundamental groups of higher-genus surfaces are more complicated, I would expect the features of solutions/functions $y$ corresponding to higher-genus algebraic curves $P(u,v)$ to be more interesting/subtle/complicated, which is why I would like a reference to learn about them.