 Nullcline

For a planar autonomous differential equation $\dot{x}=f(x,y)$, $\dot{y}=g(x,y)$, an $x$-nullcline is a set where $f=0$ and a $y$-nullcline is a set where $g=0$. Their intersections are equilibrium points. Away from intersections, the vector field is vertical on an $x$-nullcline and horizontal on a $y$-nullcline.

For $\dot{x}=x-y$, $\dot{y}=x+y$, the nullclines are $y=x$ and $y=-x$. On $y=x$ the vertical velocity is $2x$, and on $y=-x$ the horizontal velocity is $2x$. Signs in the four intervening sectors complete a qualitative phase portrait.

A nullcline need not be a trajectory. At $(1,1)$ in this example, the vertical vector $(0,2)$ crosses the diagonal nullcline rather than following it. Invariance of a curve requires tangency of the full vector field, not merely vanishing of one coordinate velocity.