  We describe all the elliptic fibrations with section on the Kummer surface X
of the Jacobian of a very general curve C of genus 2 over an algebraically
closed field of characteristic 0, modulo the automorphism group of X and the
symmetric group on the Weierstrass points of C. In particular, we compute
elliptic parameters and Weierstrass equations for the 25 different fibrations
and analyze the reducible fibers and Mordell-Weil lattices. This answers
completely a question posed by Kuwata and Shioda in 2008.
