  We consider an ensemble of phase oscillators in the thermodynamic limit,
where it is described by a kinetic equation for the phase distribution density.
We propose an ansatz for the circular moments of the distribution
(Kuramoto-Daido order parameters) that allows for an exact truncation at an
arbitrary number of modes. In the simplest case of one mode, the ansatz
coincides with that of Ott and Antonsen [Chaos 18, 037113 (2008)]. Dynamics on
the extended manifolds facilitate higher dimensional behavior such as chaos,
which we demonstrate with a simulation of a Josephson junction array. The
findings are generalized for oscillators with a Cauchy-Lorentzian distribution
of natural frequencies.
