  In the framework of the Ginzburg-Landau approach, we present a
self-consistent theory of specific soliton states in mesoscopic (thin-walled)
two-band-superconducting cylinders in external parallel magnetic fields. Such
states arise in the presence of "Josephson-type" interband coupling, when phase
winding numbers are different for each component of the superconducting order
parameter. We evaluate the Gibbs free energy of the sysyem up to second-order
terms in a certain dimensionless parameter
$\epsilon\approx\frac{\mathcal{L}_{m}}{\mathcal{L}_{k}}\ll1$, where
$\mathcal{L}_{m}$ and $\mathcal{L}_{k} $ are the magnetic and kinetic
inductance, respectively. We derive the complete set of exact soliton
solutions. These solutions are thoroughly analyzed from the viewpoint of both
local and global (thermodynamic) stability. In particular, we show that
rotational-symmetry-breaking caused by the formation of solitons gives rise to
a zero-frequency rotational mode. Although soliton states prove to be
thermodynamically metastable, the minimal energy gap between the lowest-lying
single-soliton states and thermodynamically stable zero-soliton states can be
much smaller than the magnetic Gibbs free energy of the latter states, provided
that intraband "penetration depths" differ substantially and interband coupling
is weak. The results of our investigation may apply to a wide class of
mesoscopic doubly-connected structures exhibiting two-band superconductivity.
