  It is shown that pure exponential discs in spiral galaxies are capable of
supporting slowly varying discrete global lopsided modes, which can explain the
observed features of lopsidedness in the stellar discs. Using linearized fluid
dynamical equations with the softened self-gravity and pressure of the
perturbation as the collective effect, we derive self-consistently a quadratic
eigenvalue equation for the lopsided perturbation in the galactic disc. On
solving this, we find that the ground-state mode shows the observed
characteristics of the lopsidedness in a galactic disc, namely the fractional
Fourier amplitude A$_1$ increases smoothly with the radius. These lopsided
patterns precess in the disc with a very slow pattern speed with no preferred
sense of precession. We show that the lopsided modes in the stellar disc are
long-lived because of a substantial reduction ($\sim$ a factor of 10 compared
to the local free precession rate) in the differential precession. The
numerical solution of the equations shows that the ground-state lopsided modes
are either very slowly precessing stationary normal mode oscillations of the
disc or growing modes with a slow growth rate depending on the relative
importance of the collective effect of the self-gravity. N-body simulations are
performed to test the spontaneous growth of lopsidedness in a pure stellar
disc. Both approaches are then compared and interpreted in terms of long-lived
global $m=1$ instabilities, with almost zero pattern speed.
