  In this article we focus on multiplier Hermitian-Einstein metrics introduced
by Mabuchi which include K\"ahler-Einstein metrics, K\"ahler-Ricci solitons and
Mabuchi solitons as special cases. We also focus on KSM-manifolds, which are
introduced by the first author as toric bundles, to establish a criterion for
the existence of multiplier Hermitian-Einstein metrics in terms of KSM-data. An
explicit example for a KSM-manifold admitting a family of multiplier
Hermitian-Einstein metrics is constructed by using a continuous path connecting
a K\"ahler-Ricci soliton and a Mabuchi soliton.
