  We start by surveying the planar point vortex motion of the Euler equations
in the whole plane, half-plane and quadrant. Then we go on to prove
non-collision property of 2-vortex system by using the explicit form of orbits
of 2-vortex system in the half-plane. We also prove that the $N$-vortex system
in the half-plane is nonintegrable for $N>2$, which is suggested previously by
numerical experiments without rigorous proof.
  The skew-mean-curvature (or binormal) flow in $\mathbb R^n,\;n\geq3$ with
certain symmetry can be regarded as point vortex motion of the 2D lake
equations. We compare point vortex motions of the Euler and lake equations.
Interesting similarities between the point vortex motion in the half-plane,
quadrant and the binormal motion of coaxial vortex rings, sphere product
membranes are addressed. We also raise some open questions in the paper.
