Which solvable ODE's correspond to algebraic curves? This is a follow-up to my previous question: Are there periodic functions satisfying a quadratic differential equation?
Let $P(u,v)$ be a bivariate polynomial. Then its zero set $\{(u,v): P(u,v)=0\}$ is an algebraic curve. It might make it easier to assume in what follows that the curve has no singularities.
If we substitute $y$ and $y'$ into $P(u,v)$ to take the places of $u$ and $v$ respectively, then we get a possibly non-linear first-order ODE. I am curious about how many ODEs of this type have solutions.

1. How many/which first order ODE's formed by substituting into a bivariate polynomial have solutions?
2.  Does any solution to such an ODE parametrize the algebraic curve that characterizes the defining ODE?  EDIT: I realize now that the answer to this question is obviously yes, because it follows directly from the fact that $y$ and $y'$ are both functions of the same single variable (say $t$) and that they satisfy the equation $P(y,y')=0$, so if they exist they must parametrize the algebraic curve, as a result of their definition.

Some examples might help. For second-degree algebraic curves, we have that the equation of the unit circle corresponds to a first order ODE which has both sine and cosine as solutions, and that these two functions parametrize the unit circle as well: $$(y')^2 + y^2 -1=0  $$ Likewise, the hyperbolic sine and hyperbolic cosine parametrize the unit hyperbola, and both correspond to solutions of a first order ODE whose form is that of the unit hyperbola: $$(y')^2 - y^2  -1=0$$ Finally, smooth cubic curves in Weierstrass normal form can be parametrized by the Weierstrass $\wp$ functions, and the (complex) differential equations characterizing the Weierstrass $\wp$ functions have the form of a smooth cuvic curve in Weierstrass normal form: $$(\wp')^2 = 4[\wp^2] -g_2 \wp -g_3  $$ Then my question is essentially: how far does this go? For examples, if I were to choose the algebraic curve $$ v^4 - v^3 = u^5 +u^2 -7$$ would the first-order ODE $$(y')^4 - (y')^3 = y^5 + y^2 -7$$ have a solution? And would the solution to this ODE (if it existed) parametrize the original algebraic curve $v^4 - v^3 = u^5 + u^2 -7$? This example is completely arbitrary, but hopefully it makes clear to some extent the level of generality I am interested in.
Note: I am not sure how to properly tag this question.
This question is related but addresses second order ODE's -- however, I am not interested in second order ODE's, only first order: Algebraic Curves and Second Order Differential Equations
I think that this question is probably the most related, assuming that epicycloids can be defined by first-order ODEs (I don't know either way). Proof that Epicycloids are Algebraic Curves?
 A: This isn't really so much of an answer as a compendium of relevant links I have found about the subject on the internet -- most seem to take a much more abstract approach than what I had in mind, which was just to generalize the trigonometric and Weierstrass $\wp$ functions.


*

*https://www.kent.ac.uk/smsas/personal/mgr/aadios2011/slides/Winkler.pdf

*http://www.risc.jku.at/publications/download/risc_4990/grasegger_issac14.pdf

*http://www.risc.jku.at/education/courses/ss2016/ComputerAnalysis/overview-AODEs.pdf

*http://dspace.uah.es/dspace/bitstream/handle/10017/21103/VersionRepositorioSODE_revisedversion_LastraNgoSendraWinklerV3.pdf?sequence=1

*https://www.ricam.oeaw.ac.at/specsem/specsem2013/workshop3/slides/winkler.pdf

*http://www.mmrc.iss.ac.cn/~xgao/paper/p42-feng.pdf

*http://www.mmrc.iss.ac.cn/~xgao/paper/alg-sol.pdf

*http://www.mmrc.iss.ac.cn/~dart4/posters/Ngo.pdf

*http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4608394/

*http://www.risc.jku.at/publications/download/risc_4990/grasegger_issac14.pdf
A: What you're looking for is, probably, the following theorems: (Theorems A and B of "A Malmquist–Yosida type of theorem for the
second-order algebraic differential equations"  by Liangwen Liao)

Theorem (Malmquist): Let $R(z,f)$ be birational. If a differential equation of the form $$f'(z)=R(z,f)$$ admits a transcendental meromorphic solution, then the equation can be reduced into a Riccati differential equation $$f'(z)=a(z)+b(z)f(z)+c(z)f^2(z),$$ where $a(z), b(z)$ and $c(z)$ are rational functions.

That is, if your polynomial $P(y,y')$ has degree $1$ in $y'$, it must have degree at most $2$ in $y$.

Theorem (Steinmetz): Let $R(z,f)$ be birational. If the following differential equation $$f'(z)^n=R(z,f),$$ admits a transcendental meromorphic solution, then the differential equation can be reduced into $$f'(z)^n=\sum_{i=0}^{2n}a_i(z)f(z)^i,$$ where the $a_i(z)(i=0,1,...,n)$ are rational functions and at least one of then does not vanish.

So if your equation has degree $n$ in $y'$, then it must have degree at most $2n$ in $y$.
The general case with more then on power of $y'$ is adressed in the "Fusch conditions" that you can find in section $1$ of  "Meromorphic solutions of algebraic differentlal equations" by Eremenko.
