Isolated singularities of solutions $u,v$ of (32) have the following properties:

• (a)

  Let $u,v$ satisfy (32) on an open set $U$ in $\mathbin{\mathbb{R}}^{2}$, and let $(x,0)$ be an isolated singularity of $u,v$. Then the order of $(x,0)$ is a positive integer.

• (b)

  For each $k\geqslant 1$, there exist solutions $u,v$ of (32) defined on a small ball $B$ about $(0,0)$ in $\mathbin{\mathbb{R}}^{2}$, with an isolated zero of order $k$ at $(0,0)$.

• (c)

  For $k$ odd, the solutions $u,v$ in part (b) may be chosen to satisfy

  $$u(x,y)=u(x,-y)=-u(-x,y)\quad\text{and}\quad v(x,y)=-v(x,-y)=v(-x,y)$$

  for all $(x,y)\in B$, and such that $u(x,0)$ is a strictly increasing function.

• (d)

  For $k$ even, the solutions $u,v$ in part (b) may be chosen to satisfy

  $$u(x,y)=u(x,-y)=u(-x,y)\quad\text{and}\quad v(x,y)=-v(x,-y)=-v(-x,y)$$

  for all $(x,y)\in B$, and such that $u(x,0)$ is strictly increasing for $x>0$ and strictly decreasing for $x<0$.