  We use the Hubbard Hamiltonian $H$ on the honeycomb lattice to represent the
valence bands of carbon single-wall $(N,N)$ nanotubes. A detailed symmetry
analysis shows that the model allows W=0 pairs which we define as two-body
singlet eigenstates of $H$ with vanishing on-site repulsion. By means of a
non-perturbative canonical transformation we calculate the effective
interaction between the electrons of a W=0 pair added to the interacting ground
state. We show that the dressed W=0 pair is a bound state for resonable
parameter values away from half filling. Exact diagonalization results for the
(1,1) nanotube confirm the expectations. For $(N,N)$ nanotubes of length $l$,
the binding energy of the pair depends strongly on the filling and decreases
towards a small but nonzero value as $l \to \infty$. We observe the existence
of an optimal doping when the number of electrons per C atom is in the range
1.2$\div$1.3, and the binding energy is of the order of 0.1 $\div$ 1 meV.
