Introducing a new notion of algebraic integrability

Let us consider the general homogeneous quadratic dynamic system. We will call it algebraically integrable by given functions $h_{1},\dots ,h_{n}$ if the set of roots of the equation $\xi ^{n}-h_{1}\xi ^{n-1}+\dots +(-1)^{n}h_{n}\equiv 0$ solves the dynamic system.
The talk introduces this new notion of algebraic integrability and presents a wide class of quadratic dynamic systems that are algebraically integrable by the set of functions $h_{1},\dots ,h_{n}$ where $h_{1}$ is the solution to an ordinary differential equation of order $n$ and $h_{k}$ are differential polynomials in $h_{1}$, $k=2,\dots ,n$. Results on algebraically integrable quadratic dynamic systems and non-linear ordinary differential equations related to them are obtained. Classical examples like the Darboux–Halphen system are considered.