= Cubic planar flow with diagonal approach directions
{title2=$\dot x=x(x^2+y^2),\qquad\dot y=2y^3$}
For this cubic <vector field>, every non-axis trajectory has the <first integral> $I(x,y)=1/y-y/x^2$. Direct differentiation gives $\nabla I\cdot(\dot x,\dot y)=0$ where $xy\ne0$. Equivalently the trajectories satisfy $x^2-y^2=Cx^2y$. The axes are separate invariant trajectories, and the diagonals are the cases $C=0$. For each fixed finite $C$, approach to the origin forces $y^2/x^2=1-Cy\to1$, so non-axis trajectories have diagonal limiting directions.
The origin is repelling in forward time. For $V=x^2+y^2$, $\dot V=2x^4+2x^2y^2+4y^4\geq V^2$, so every nonzero trajectory eventually exits any small disk. The zero <Jacobian matrix> at the origin leaves the linearized system constant, demonstrating why nonlinear terms are needed for this stability classification.
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