  Answering the question of V.I. Oseledets, we present a random variable $\xi$
such that the sum $\xi(x)+a\xi(y)$ has a singular distribution for a set of
parameters $a$ dense in $(1, +\infty)$, but for another dense set of
parameters, this sum has an absolutely continuous distribution. We prove the
following assertion: given $C,D$, countable non-intersecting dense subsets of
the ray $(1,+\infty)$, there is a measure-preserving flow $T_t$ (acting on the
infinite Lebesgue space) such that automorphisms $T_1\otimes T_{c}$ have simple
singular spectra for every $c\in C$, and $T_1\otimes T_{d}$ have Lebesgue
spectra for all $d\in D$. The spectral measure of this flow plays the role of
the distribution of our random variable $\xi$.
