  The definition of balanced metrics was originally given by Donaldson in the
case of a compact polarized K\"{a}hler manifold in 2001, who also established
the existence of such metrics on any compact projective K\"{a}hler manifold
with constant scalar curvature. Currently, the only noncompact manifolds on
which balanced metrics are known to exist are homogeneous domains. The
generalized Cartan-Hartogs domain
$\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu)$ is defined as the
Hartogs type domain constructed over the product $\prod_{j=1}^k\Omega_j$ of
irreducible bounded symmetric domains $\Omega_j$ $(1\leq j \leq k)$, with the
fiber over each point $(z_1,...,z_k)\in \prod_{j=1}^k\Omega_j$ being a ball in
$\mathbb{C}^{d_0}$ of the radius
$\prod_{j=1}^kN_{\Omega_j}(z_j,\bar{z_j})^{\frac{\mu_j}{2}}$ of the product of
positive powers of their generic norms. Any such domain
$\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu)$ $(k\geq 2)$ is a
bounded nonhomogeneous domain. The purpose of this paper is to obtain necessary
and sufficient conditions for the metric $\alpha g(\mu)$ $(\alpha>0)$ on the
domain $\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu)$ to be a
balanced metric, where $g(\mu)$ is its canonical metric. As the main
contribution of this paper, we obtain the existence of balanced metrics for a
class of such bounded nonhomogeneous domains.
