  The classical problem of the Josephson junction of arbitrary length W in the
presence of externally applied magnetic fields (H) and transport currents (J)
is reconsidered from the point of view of stability theory. In particular, we
derive the complete infinite set of exact analytical solutions for the phase
difference that describe the current-carrying states of the junction with
arbitrary W and an arbitrary mode of the injection of J. These solutions are
parameterized by two natural parameters: the constants of integration. The
boundaries of their stability regions in the parametric plane are determined by
a corresponding infinite set of exact functional equations. Being mapped to the
physical plane (H,J), these boundaries yield the dependence of the critical
transport current Jc on H. Contrary to a wide-spread belief, the exact
analytical dependence Jc=Jc(H) proves to be multivalued even for arbitrarily
small W. What is more, the exact solution reveals the existence of unquantized
Josephson vortices carrying fractional flux and located near one of the
junction edges, provided that J is sufficiently close to Jc for certain finite
values of H. This conclusion (as well as other exact analytical results) is
illustrated by a graphical analysis of typical cases.
