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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

Ali Taghavi
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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
Igor Rivin
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