  We study left invariant conformal Killing-Yano (CKY) $2$-forms on Lie groups
endowed with a left invariant metric. The first examples of CKY $2$-forms on
metric Lie algebras which do not admit any Sasakian structure, are exhibited.
We classify, up to isometry and scaling, $5$-dimensional metric Lie algebras
with center of dimension greater than one admitting CKY tensors. In addition,
we construct $5$-dimensional metric Lie algebras with one-dimensional center
carrying CKY tensors. In both cases, we determine all possible CKY tensors on
these metric Lie algebras.
