  Let f be a class P -homeomorphism of the circle. We prove that there exists a
piecewise analytic homeomorphism that conjugate f to a one-class P with
prescribed break points lying on pairwise distinct orbits. As a consequence, we
give a sharp estimate for the smoothness of a conjugation of class P
-homeomorphism f of the circle satisfying the (D)-property (i.e. the product of
f-jumps in the break points contained in a same orbit is trivial), to
diffeomorphism. When f does not satisfy the (D)-property the conjugating
homeomorphism is never piecewise C^1 and even more it is not absolutely
continuous function if the total product of f-jumps in all the break points is
non-trivial.
