  In this paper, we show that $G$-invariant Calabi-Yau structures on the
complexification $G^{\mathbb C}/K^{\mathbb C}$ of a symmetric space $G/K$ of
compact type are constructed from solutions of a Monge-Amp$\grave{\rm e}$re
type equation. Also, we give an explicit descriptions of the
Monge-Amp$\grave{\rm e}$re type equation in the case where the rank of $G/K$ is
equal to one or two. Furthermore, we prove the existence of solutions of the
Monge-Amp$\grave{\rm e}$re type equation.
