What is a complete description for the configuration of zero locus of the algebraic curve $C$ defined by $$yP(x,y)-xQ(x,y)=0$$


where $P,Q \in \mathbb{R}[x,y]$ are arbitrary polynomials of degree $2$.


What is the (sharp) maximum number of connected components of $\mathbb{R}^2 \setminus C$?


This question is motivated by "**EDITED**" part of the following answer:
[Finding a 1-form adapted to a smooth flow](https://mathoverflow.net/questions/273635/finding-a-1-form-adapted-to-a-smooth-flow/273648#273648)



The configuration of the zero locus of certain polynomial
Your curve is a planar cubic, and such were classified completely recently by one Isaac Newton. An even more recent version can be found in this 1928 Annals of Math article:



```
Canonical Forms of Plane Cubic Curves Under Euclidean Transformations
R. S. Burington and H. K. Holt
Annals of Mathematics
Second Series, Vol. 30, No. 1/4 (1928 - 1929), pp. 52-60

```

