Geodesic webs and integrable geodesic flows

For Liouville integrability of a geodesic flow on a surface, it is enough to find one integral independent of the Hamiltonian.
Using an integral, polynomial and homogeneous in momenta of degree three, we construct a geodesic 3-web. This web turns out to be hexagonal. Conversely, existence of a hexagonal geodesic 3-web implies existence of a cubic integral.
Using a quadratic integral, we construct a one-parameter family of geodetic nets (2-webs). Integrating the bisector direction fields for each such net, we obtain a 4-web with hexagonal 3-subwebs. Conversely, existence of a geodesic net with this property implies existence of a quadratic integral.
Using an integral, fractional-linear in momenta, we construct a one-parameter family of geodesic foliations. Fixing 4 values of the parameter, we obtain a geodesic 4-web with constant cross-ratio of its tangent directions. Conversely, existence of such a 4-web implies existence of a fractional linear-integral.
For a geodesic flow to be integrable on a three-dimensional manifold, existence of two integrals in involution is necessary. A famous example with two quadratic integrals was constructed by Stäckel at the end of the 19th century. It turns out that the corresponding metric and integrals are described in terms of webs of maximal rank.