  A modified interaction representation for the master field describing
connected $SU(N)$-invariant Wightman's functions in the large $N$ limit of
matrix fields is constructed. This construction is based on the representation
of the master field in terms of Boltzmannian field theory found before. In the
modified interaction representation we deal with two scalar Boltzmann fields
({\it up} and {\it down} fields). For up and down fields only half-planar
diagrams contribute and this could help to write down a recursive set of
non-linear integral-differential equations summing up planar diagrams.
