  We derive explicit semiclassical quantisation conditions for the Dirac and
Pauli equations. We show that the spin degree of freedom yields a contribution
which is of the same order of magnitude as the Maslov correction in
Einstein-Brillouin-Keller quantisation. In order to obtain this result a
generalisation of the notion of integrability for a certain skew product flow
of classical translational dynamics and classical spin precession has to be
derived. Among the examples discussed is the relativistic Kepler problem with
Thomas precession, whose treatment sheds some light on the amazing success of
Sommerfeld's theory of fine structure [Ann. Phys. (Leipzig) 51 (1916) 1--91].
