  Belavin's $\mathbb{Z}_n$-symmetric elliptic model with boundary reflection is
considered on the basis of the boundary CTM bootstrap. We find non-diagonal
$K$-matrices for $n>2$ that satisfy the reflection equation (boundary
Yang--Baxter equation), and also find non-diagonal Boltzmann weights for the
$A^{(1)}_{n-1}$-face model even for $n\geqq 2$. We derive difference equations
of the quantum Knizhnik-Zamolodchikov type for correlation functions of the
boundary model. The boundary spontaneous polarization is obtained by solving
the simplest difference equations. The resulting quantity is the square of the
spontaneous polarization for the bulk $\mathbb{Z}_n$-symmetric model, up to a
phase factor.
