Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes

We present a model of set theory, in which, for a given n≥2, there exists a non-ROD-uniformizable planar lightface ¹ₙ set in R× R, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface Σ¹ₙ sets with countable cross-sections are Δ¹ₙ₊₁-uniformizable. Thus it is true in this model, that the ROD-uniformization principle for sets with countable cross-sections first fails precisely at a given projective level.