  In the paper $k$-multiple self-intersection local time for planar Gaussian
integrators generated by linear operator with nontrivial kernel is studied. In
this case additional singularities arise in its formal Fourier--Wiener
transform. In case $k=2$ the set of singularities is the finite number of
points. In case $k>2$ it contains intervals and hyperplanes. In the first and
the second cases using two different approaches related on structure of set of
singularities we show that "new" singularities do not imply on the convergence
of integral corresponding to the formal Fourier--Wiener transform and
regularization consist of compensation of impact of diagonals as for the Wiener
process.
