  We prove that a one-dimensional foliation with generic singularities on a
projective space, exhibiting a Lie group transverse structure in the complement
of some codimension one algebraic subset is logarithmic, i.e., it is the
intersection of codimension one foliations given by closed one-forms with
simple poles. If there is only one singularity in a suitable affine space, then
the foliation is induced by a linear diagonal vector field.
