Algebraic anzatz for heat equation and integrable polynomial dynamical systems

We will discuss the ansatz that reduces the heat equation to a homogeneous polynomial dynamical system. For any such system in the generic case we obtain a nonlinear ordinary differential equation and algorithm for constructing a solution of this system. As result we have the corresponding solution of the heat equation. We give the full classification of nonlinear ordinary differential equations that arise from our ansatz.
The talk is based on recent joint works with E. Yu. Bunkova. Main definitions will be given during the talk.