Kakutani equivalence for products of some special flows over rotations

We study Kakutani equivalence for products of some special flows over rotations with roof function smooth except a singularity at $0\in \mathbb{T}$. We estimate the Kakutani invariant for product of these flows with different powers of singularities and rotations from a full measure set. As a corollary, we obtain a countable family of pairwise non-Kakutani equivalent products of special flows over rotations.