 Geodesics of the product-weighted diagonal plane metric

On a quadrant $xy\ne 0$, the geodesic Lagrangian gives $y^2\ddot{x}+2y\dot{y}\dot{x}-x{\dot{y}}^2=0$ and $x^2\ddot{y}+2x\dot{x}\dot{y}-y{\dot{x}}^2=0$. Rays $y=mx$ reduce these to $(x\dot{x}{)}^{\prime }=0$ and hence $x^2=a\lambda +b$. The coordinate change $\rho =|xy|/\sqrt{2}$, $\vartheta =\log |y/x|$ gives the locally flat metric tensor $d{\rho }^2+{\rho }^{2}d{\vartheta }^2$. Here $\vartheta$ ranges over the real line, so the polar-plane representation is local, rather than a globally one-to-one map onto the punctured plane. The axes are excluded by the degenerate metric tensor; these rays do not establish continuation through the origin as a regular geodesic.