Absolute continuity and singularity of spectra for the flows $T_t\otimes T_{at}$

Given disjoint countable dense subsets $C$ and $D$ of the half-line $(1,+\infty)$, there exists a flow $T_t$ preserving a sigma-finite measure and such that all automorphisms $T_1\otimes T_{c}$ with $c\in C$ have simple singular spectrum and all automorphisms $T_1\otimes T_{d}$ with $d\in D$ have Lebesgue spectrum of countable multiplicity.