This problem comes from Chicone's book (exercise 1.25). It asks to find the flow (i.e. the solution of the IVP) of the following system of ODEs:
\begin{align*} \dot{x} &= y^{2}-x^{2} \\ \dot{y} &= -2xy \end{align*}
A hint is given to consider the complex variable $z = x+iy$, but I cannot quite seem to figure out a solution (it is not separable or Hamiltonian). Numerically, it looks like most solutions lie on a circle containing the origin, and this seems to be hinted at in the question, but I do not see the equation; I tried looking for a quadratic conserved quantity (to reduce the dimensionality of the system) of the form \begin{align*} V(x,y) &= ax^{2} + by^{2} + cxy + d x + ey + f, \end{align*} but none seems to exist.
Does anyone have any suggestions on how to proceed?