  We construct exactly solvable models of twisted carbon nanotubes via
supersymmetry, by applying the matrix Darboux transformation. We derive the
Green's function for these systems and compute the local density of states.
Explicit examples of twisted carbon nanotubes are produced, where the
back-scattering is suppressed and bound states are present. We find that the
local density of states decreases in the regions where the bound states are
localized. Dependence of bound-state energies on the asymptotic twist of the
nanotubes is determined. We also show that each of the constructed unextended
first order matrix systems possesses a proper nonlinear hidden supersymmetric
structure with a nontrivial grading operator.
