  We study the linear perturbations of collisionless near-Keplerian discs. Such
systems are models for debris discs around stars and the stellar discs
surrounding supermassive black holes at the centres of galaxies. Using a
finite-element method, we solve the linearized collisionless Boltzmann equation
and Poisson's equation for a wide range of disc masses and rms orbital
eccentricities to obtain the eigenfrequencies and shapes of normal modes. We
find that these discs can support large-scale `slow' modes, in which the
frequency is proportional to the disc mass. Slow modes are present for
arbitrarily small disc mass so long as the self-gravity of the disc is the
dominant source of apsidal precession. We find that slow modes are of two
general types: parent modes and hybrid child modes, the latter arising from
resonant interactions between parent modes and singular van Kampen modes. The
most prominent slow modes have azimuthal wavenumbers $m=1$ and $m=2$. We
illustrate how slow modes in debris discs are excited during a fly-by of a
neighbouring star. Many of the non-axisymmetric features seen in debris discs
(clumps, eccentricity, spiral waves) that are commonly attributed to planets
could instead arise from slow modes; the two hypotheses can be distinguished by
long-term measurements of the pattern speed of the features.
