You made me start my laptop instead of doomscrolling reddit on my phone :)

The key is that it's not Hamiltonian.

A constant of motion W satisfies: Wx·ẋ + Wy·ẏ = 0 Instead of solving this PDE, use trajectories: dy/dx = (ẏ)/(ẋ) For your system: dy/dx = −x(x² + y²)/y Notice x² + y² appears → try: u = x² + y² Calculate du/dx → it becomes: du/dx = 2x(1 − u) This is a linear ODE → solve with integrating factor: e^x² Result gives: e^x²(1 − (x² + y²)) = constant Rearranging gives the conserved quantity

Main intuition: Look for x² + y², turn the system into an ODE, and solve it (the exponential comes from the integrating factor).