  We analyze mean-field models of synchronization of phase oscillators with
singular couplings and subject to external random forces. They are related to
the Kuramoto-Sakaguchi model. Their probability densities satisfy local partial
differential equations similar to the Porous Medium, Burgers and extended
Burgers equations depending on the degree of singularity of the coupling. We
show that Porous Medium oscillators (the most singularly coupled) do not
synchronize and that (transient) synchronization is possible only at zero
temperature for Burgers oscillators. The extended Burgers oscillators have a
nonlocal coupling first introduced by Daido and they may synchronize at any
temperature. Exact expressions for their synchronized phases and for Daido's
order function are given in terms of elliptic functions.
