  We present idealized models of a razor-thin, axisymmetric, Keplerian stellar
disc around a massive black hole, and study non-axisymmetric secular
instabilities in the absence of either counter-rotation or loss cones. These
discs are prograde mono-energetic waterbags, whose phase space distribution
functions are constant for orbits within a range of eccentricities (e) and zero
outside this range. The linear normal modes of waterbags are composed of
sinusoidal disturbances of the edges of distribution function in phase space.
Waterbags which include circular orbits (polarcaps) have one stable linear
normal mode for each azimuthal wavenumber m. The m = 1 mode always has positive
pattern speed and, for polarcaps consisting of orbits with e < 0.9428, only the
m = 1 mode has positive pattern speed. Waterbags excluding circular orbits
(bands) have two linear normal modes for each m, which can be stable or
unstable. We derive analytical expressions for the instability condition,
pattern speeds, growth rates and normal mode structure. Narrow bands are
unstable to modes with a wide range in m. Numerical simulations confirm linear
theory and follow the non-linear evolution of instabilities. Long-time
integration suggests that instabilities of different m grow, interact
non-linearly and relax collisionlessly to a coarse-grained equilibrium with a
wide range of eccentricities.
