  For a non-compact simple Lie algebra $\mathfrak{g}$ over $\mathbb{R}$, we
denote by $\mathcal{O}^{\mathbb{C}}_{\min,\mathfrak{g}}$ the unique complex
nilpotent orbit in $\mathfrak{g} \otimes_\mathbb{R} \mathbb{C}$ containing all
minimal real nilpotent orbits in $\mathfrak{g}$. In this paper, we give a
complete classification of symmetric pairs $(\mathfrak{g},\mathfrak{h})$ such
that $\mathcal{O}^{\mathbb{C}}_{\min,\mathfrak{g}} \cap \mathfrak{g}^d =
\emptyset$, where $\mathfrak{g}^d$ denotes the dual Lie algebra of
$(\mathfrak{g},\mathfrak{h})$. Furthermore, for symmetric pairs $(G,H)$ with
real simple Lie group $G$, we apply our classification to theorems given by T.
Kobayashi [J. Lie Theory (2023)], and study bounded multiplicity properties of
restrictions on $H$ of infinite-dimensional irreducible $G$-representations
with minimum Gelfand--Kirillov dimension.
