Foliations connected with the Monge–Ampère equation in Hartogs domains

Let $D$ be a domain in $\mathbf C^2(z,w)$, and $u$ a solution in $D$ of the equation $(\partial\overline\partial u)^2=0$, where $\partial\overline\partial u\ne0$ in $D$. It is known that, for $u\in C^3(D)$, $D$ is foliated into complex curves on which $u$ is harmonic, and $\partial u/\partial z$ and $\partial u/\partial w$ are holomorphic. We show that if $u=u(|z|,w)$ and $D$ is a complete Hartogs domain with axis of symmetry $z=0$, then such a foliation exists even for $u\in C^2(D)$.
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