  For positive functions $u\in C^{2}(\Omega) $, where $\Omega$ is an open
subset of $\mathbb{R}^{n}$, the Symmetric Minimal Surface Equation (SME), is
$\sum_{i=1}^{n}D_{i}\bigl(\frac{D_{i}u}{\sqrt{1+|Du|^{2}}}\bigr)=\frac{m-1}{u\sqrt{1+|Du|^{2}}}$.
Geometrically, the SME expresses the fact that the ``symmetric graph'' $SG(u)$,
defined by $SG(u)=\bigl\{(x,\xi)\in
\Omega\times\mathbb{R}^{m}:|\xi|=u(x)\bigr\}$, is a minimal (i.e.\ zero mean
curvature) hypersurface in $\Omega\times\mathbb{R}^{m}$. A function $u\in
C^{1}(\Omega)$ is said to be a singular solution if $u^{-1}\{0\}\neq
\emptyset$, and if $u=\lim_{j\to\infty}u_{j}$, uniformly on each compact subset
of $\Omega$, where each $u_{j}$ is a positive $C^{2}(\Omega)$ solution of the
SME. The present paper develops are theory of singular solutions of the SME,
including existence, H\"older and Lipschitz estimates for bounded solutions,
and a compactness and regularity theory. We also prove that the singular set
$u^{-1}{\{0\}}$ is codimension at most 2.
