In this paper, we are interested in the following degenerate elliptic Monge-Ampère equation: ... Under suitable structure conditions on f(t), we can show that and the solutions of linearized equation have the same symmetric property as the domain. Moreover, we also achieve some uniqueness results for homogenous nonlinearity f(t) = tp (p > n) in general convex domain. Compared to the results in [G. Huang, Calc. Var. Partial Differential Equations, 58 (2019), 73], we dispose the "uniformly convex" condition imposed on the domain. With a subtle approximation procedure and variant form of Hopf's lemma, we overcome the technical difficulties caused by the degeneracy of the above equation. What's more, we can also get the uniqueness property for domains whose geometry is close enough to the symmetric convex domain. [ABSTRACT FROM AUTHOR]