  Let $u$ denote a solution to a rotationally invariant Hessian equation
$F(D^2u)=0$ on a bounded simply connected domain $\Omega\subset R^2$, with
constant Dirichlet and Neumann data on $\partial \Omega$. In this paper we
prove that if $u$ is real analytic and not identically zero, then $u$ is radial
and $\Omega$ is a disk. The fully nonlinear operator $F\not\equiv 0$ is of
general type, and in particular, not assumed to be elliptic. We also show that
the result is sharp, in the sense that it is not true if $\Omega$ is not simply
connected, or if $u$ is $C^{\infty}$ but not real analytic.
