  In this paper, we study symmetry and existence of solutions of minimal
gradient graph equations on punctured space $\mathbb R^n\setminus\{0\}$, which
include the Monge-Amp\`ere equation, inverse harmonic Hessian equation and the
special Lagrangian equation. This extends the classification results of
Monge-Amp\`ere equations. Under some conditions, we also give the
characterization of the solvability on exterior Dirichlet problem in terms of
their asymptotic behaviors.
