  Recently, the definitions of the Kimber-Martin-Ryskin-Watt (KMRW)
unintegrated parton densities (UPDFs) have been discussed by several groups. In
the first part of this manuscript, we remind the issues encountered with these
definitions and discuss the proposed solutions. In our opinion, none of these
solutions is fully satisfactory. We observe that these issues seem to be
related to the normalization condition where the UPDFs are related to the
collinear PDFs by an integration over transverse momentum cut off by the
factorization scale. Then, we build a modified version of the angular-ordering
KMRW UPDFs, obeying the normalization condition with the transverse momentum
integrated up to infinity and show that the usual issues are absent.
