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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 cyclicautomorphism $h_M:H_M\to H_M$ whose eigenvalues are the unit roots with ordersin $M$ 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 andsufficient 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 cyclotomicpolynomials. It is elementary, but involved. Special cases of the main resulthave been applied to the study of the automorphisms of Milnor lattices ofisolated hypersurface singularities.

Automorphisms with eigenvalues in $S^1$ of a ${\mathbb Z}$-lattice with cyclic finite monodromy