  In the paper we study characterizations of probability measures in free
probability. By constancy of regressions for random variable $\V(\I-\U)\V$
given by $\V\U\V$, where $\U$ and $\V$ are free, we characterize free Poisson
and free binomial distributions. Our paper is analog in free probability of
results known in classical probability \cite{BobWes2002Dual}, where gamma and
beta distributions are characterized by constancy of $\E((V(1-U))^{i}|UV)$, for
$i \in \{-2,-1,1,2\}$. This paper together with previous results
\cite{SzpojanWesol} exhaust all cases of characterizations from
\cite{BobWes2002Dual}.
