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We calculate the anisotropic two-photon absorption spectra of an AlGaAs-GaAs V-shaped quantum wire with realistic band structure, including excitonic effects. The absorption coefficient is obtained by integrating effective multiband Bloch equations accounting for the Coulomb interaction within the Hartree-Fock approximation. The various excitonic peaks are identified with respect to the involved energy subbands and the symmetry properties. Furthermore, A2-excitons, which are dark for one-photon excitation, become bright for two-photon spectroscopy when the light is linearly polarized along a direction that is not a symmetry axis. We show that some of them may be observable.