It is well known that the spectrum of dynamical systems arising from generalized Morse sequences (M. Keane. Generalized Morse sequences. Z. Wahr. Verw. Geb.10 (1968), 335–353.) is simple as soon as it is non-discrete. We give, in this paper, a necessary and sufficient condition for the existence of such a transformation with non-purely singular spectrum. From this, it follows that this problem is equivalent to an open problem of the existence of ‘flat’ polynomials on the Torus group. We show that this latter question can be given an affirmative answer on some other group, and this allows us to construct a countable abelian group action with simple spectrum whose spectral type is the sum of a discrete measure and of the Haar measure on the dual group.