  We construct rigid Poisson suspensions without roots. The discrete rational
component in spectrum of an ergodic automorphism S prevents some roots from
existing. If S is tensorly multiplied by an ergodic automorphism of the space
with a sigma-finite measure, discrete spectrum disappears in this product, but
like the smile of Cheshire Cat, the memory of it can remain in the form of the
absence of roots. In additional conditions, this effect is inherited by the
Poisson suspension over the above product. Starting from this idea, but without
using the tensor product, we describe simple rank-one constructions for which
the Poisson suspensions are rigid and have no roots.
