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 system up to second-order terms in a certain dimensionless parameter <$E epsilon~symbol Ы~L sub m "/" L sub k~<< <<~1>, where <$E L sub m> and <$E L sub 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.

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