Rigid Poisson suspensions without roots

The presence of a discrete rational component in the spectrum of an ergodic automorphism $S$ is inconsistent with the existence of certain roots of $S$. If $T$ is an ergodic automorphism of a space with $\sigma$-finite measure, then the discrete spectrum disappears from the product $S\otimes T$, but the memory of it may remain in the form of the absence of roots, like Cheshire Cat's grin. Under certain additional assumptions, this effect is inherited by the Poisson suspension over such a product. Based on this idea, we propose a simple rank-one construction for which the Poisson suspension has no roots and is rigid.