  The low-energy electronic Hamiltonian for the thinest zigzag carbon nanotube,
embedded into a dielectric host, is derived and its phase diagram is discussed.
The specific multi-band structure and the microscopic form of the
electron-electron interaction in this systems is considered. The interband
repulsive interaction, which is almost unscreened, leads to a retarded
intraband attraction between the electrons and can stabilize the Cooper pairs.
For a dielectric constant of the host $\epsilon_d\sim 2-4$, the theory predicts
that the superconducting fluctuations should develop, which is in agreement
with the experiment. For $\epsilon_d \gtrsim 8$, the density wave fluctuations
should be amplified. Between the two phases, there is a metallic state where
all two-particle fluctuations are suppressed.
