  For any finite set $M\subset {\mathbb Z}_{\geq 1}$ of positive integers,
there is up to isomorphism a unique ${\mathbb Z}$-lattice $H_M$ with a cyclic
automorphism $h_M:H_M\to H_M$ whose eigenvalues are the unit roots with orders
in $M$ and have multiplicity 1. The paper studies the automorphisms of the pair
$(H_M,h_M)$ which have eigenvalues in $S^1$. The main result are necessary and
sufficient conditions on the set $M$ such that the only such automorphisms are
$\pm h_M^k,k\in{\mathbb Z}$. The proof uses resultants and cyclotomic
polynomials. It is elementary, but involved. Special cases of the main result
have been applied to the study of the automorphisms of Milnor lattices of
isolated hypersurface singularities.
