Generic planar algebraic vector fields are strongly minimal and disintegrated (Q2077143)

In the article under review, the author studies the geometry of a strongly minimal \(2\)-dimensional subset in a differentially closed field given by the equation \[ \delta(x)=f(x,y) \quad \& \quad \delta(y)=g(x,y), \] for two polynomials \(f\) and \(g\) of degree at least \(3\) with complex coefficients. He shows that if the coefficients are algebraically independent over \(\mathbb Q\), then the geometry is trivial (or disintegrated) (Theorem A). He furthermore deduces (Corollary B) that, whenever any two distinct solutions are independent, then so are any \(n\) many distinct solutions, which in particular yields a very precise description of the possible algebraic relations among solutions. It is worth noting that Corollary B has been extended in recent work of \textit{J. Freitag} et al. [Invent. Math. 230, No. 3, 1249--1265 (2022; Zbl 1506.03085)] to deduce the minimality (and desintegration) of any stationary non-algebraic type defined over the constants satisfying that any two distinct realisations are independent. The methods used by the author in the article under review reduce the original question to showing, thanks to the trichotomy for strongly minimal sets in differentially closed fields, that the generic solution of the above equation is orthogonal to the constants and that it admits, not even up to finite covers, no dominant maps to an algebraic \(1\)-dimensional \(D\)-variety. This builds on existing work by the author on the classification of the corresponding foliations.