  In this note, we construct new examples of Lorentzian Sasaki-Einstein (LSE)
metrics on Smale manifolds $M.$ It has already been established in \cite{Gmz2}
that such metrics exist on the so-called torsion free Smale manifolds, i.e. the
$k$-fold connected sum of $S^{2}\times S^{3}.$ Now, we show that LSE metrics
exist on Smale manifolds for which $H_{2}(M,\mathbb{Z})_{tor}$ is nontrivial.
In particular, we show that most simply-connected positive Sasakian rational
homology 5-spheres are also negative Sasakian (hence Lorentzian
Sasaki-Einstein). Moreover, we show that for each pair of positive integers
$(n,s)$ with $n,s >1$, there exists a Lorentzian Sasaki-Einstein Smale manifold
$M$ such that $H_{2}(M,{\mathbb{Z}})_{tors}=(\mathbb{Z}/n)^{2s}$. Finally, we
are able to construct so-called mixed Smale manifolds (connect sum of torsion
free Smale manifolds with rational homology spheres) which admit LSE metrics
and have arbitrary second Betti number. This gives infinitely many examples
which do not admit positive Sasakian structures. These results partially
address the open problems
