A properly measurable set ${\cal P} \subset \mb{X} \times M_1(\mb{Y})$ (where $\mb{X}, \mb{Y}$ are Polish spaces and $M_1(\mb Y)$ is the space of Borel probability measures on $\mb Y$) is considered. Given a probability distribution $\lambda \in M_1(\mb X)$ the paper treats the problem of the existence of $\mb{X}\times\mb{Y}$-valued random vector $(\xi,\eta)$ for which ${\cal L}(\xi)=\lambda$ and ${\cal L}(\eta | \xi=x) \in {\cal P}_x$ $\lambda$-almost surely that possesses moreover some other properties such as "${\cal L}(\xi,\eta)$ has the maximal possible support'' or "${\cal L}(\eta | \xi=x)$'s are extremal measures in ${\cal P}_x$'s''. The paper continues the research started in [7].