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Let G be a non-trivial finite group, S  G \ {e} be a set such that if a 2 S,  then a-1  S and e be the identity element of G. Suppose that Cay(G, S) is  the Cayley graph with the vertex set G such that two vertices a and b are  adjacent whenever a-1  S. An arbitrary graph is called integral whenever  all eigenvalues of the adjacency matrix are integers. We say that a group G  is Cayley integral simple whenever every connected integral Cayley graph on  G is isomorphic to a complete multipartite graph. In this paper we prove  that if G is a non-simple group, then G is Cayley integral simple if and  only if G  Zp2 for some prime number p or G  Z2 x Z2. Moreover, we show  that there exist finite non-abelian simple groups which are not Cayley  integral simple.

On groups admitting no integral Cayley graphs besides complete multipartite graphs