  We discuss an analog of the Givental group action for the space of solutions
of the commutativity equation. There are equivalent formulations in terms of
cohomology classes on the Losev-Manin compactifications of genus 0 moduli
spaces; in terms of linear algebra in the space of Laurent series; in terms of
differential operators acting on Gromov-Witten potentials; and in terms of
multi-component KP tau-functions. The last approach is equivalent to the
Losev-Polyubin classification that was obtained via dressing transformations
technique.
