  A finite non-abelian group $G$ is called super integral if the spectrum,
Laplacian spectrum and signless Laplacian spectrum of its commuting graph
contain only integers. In this paper, we first compute various spectra of
several families of finite non-abelian groups and conclude that those groups
are super integral. As an application of our results we obtain some positive
integers $n$ such that $n$-centralizer groups are super integral. We also
obtain some positive rational numbers $r$ such that $G$ is super integral if it
has commutativity degree $r$. In the last section, we show that $G$ is super
integral if $G$ is not isomorphic to $S_4$ and its commuting graph is planar.
We conclude the paper showing that $G$ is super integral if its commuting graph
is toroidal.
