Near-IR images of the prototype LINER/Seyfert type 1 galaxy NGC 1097 observed with the Very Large Telescope using adaptive optics disclose with unprecedented detail a complex central network of filamentary structure spiraling down to the center of the galaxy. The structure, consisting of several spiral arms, some almost completing a revolution about the center, is most prominent within a radius of about 300 pc. Gas and dust may be channelled to the center of NGC 1097 along this central spiral. Some filaments can be traced farther out, where they seem to connect with the nuclear star-forming ring at a 0.7 kpc radius. Straight principal shocks running along the primary large-scale bar of NGC 1097, seen in the optical images as prominent dust lanes, curve into this ring, but radio polarization vectors cross the nuclear ring at a rather large angle. Here we attempt to explain this morphology in terms of three-dimensional gas flow in a barred galaxy. In our scenario, parts of the principal shock that propagate in the off-plane gas can cross the nuclear star-forming ring and excite waves inward from it. If the dispersion relation of the excited waves allows for their propagation, they will naturally take the shape of the observed central spiral. The nuclear region of NGC 1097 remains unresolved at subarcsecond scales in the near-IR, with an upper size limit of <10 pc FWHM. Thus, any putative central dusty torus or gaseous disk envisaged by the active galactic nucleus (AGN) unified schemes has to be smaller than 10 pc in diameter at near-IR wavelengths. The extinction in the region between the nuclear star-forming ring and the nucleus increases very moderately, reaching Av~1 at the immediate surroundings of the nucleus. Thus, if the nuclear filaments are tracing cold dust, they contribute to a very low extinction in the line of sight and are likely to be distributed in a rather thin disk.