  In this paper, we provide new examples of Levi-Civita Ricci-flat Hermitian
metrics on certain compact non-K\"{a}hler Calabi-Yau manifolds, including every
compact Hermitian Weyl-Einstein manifold, every compact locally conformal
hyperK\"{a}hler manifold, certain suspensions of Brieskorn manifolds, and every
generalized Hopf manifold provided by suspensions of exotic spheres. These
examples generalize previous constructions on Hopf manifolds. Additionally, we
also construct new examples of compact Hermitian manifolds with nonnegative
first Chern class that admit constant strictly negative Riemannian scalar
curvature. Further, we remark some applications of our main results in the
study of the Chern-Ricci flow on compact Hermitian Weyl-Einstein manifolds. In
particular, we describe the Gromov-Hausdorff limit for certain explicit
finite-time collapsing solutions which generalize previous constructions on
Hopf manifolds.
