  In this paper, we analyze the effect which the choice of a friction model has
on tippe top inversion in the case where the resulting action of all
dissipative forces is described not only by the force applied at the contact
point, but also by the additional rolling resistance torque. We show that,
depending on the friction model used, the system admits different first
integrals. In particular, we give examples of friction models where the Jellett
integral, the Lagrange integral or the area integral is preserved.
  We examine in detail the case where the action of all dissipative forces
reduces to the horizontal rolling resistance torque. For this case we find
permanent rotations of the system and analyze their linear stability. Also, we
show that for this friction model no inversion is observed.
