Solution Transformations for GS String in AdS₅ x S⁵ by Conserved Quantities

For Light-cone gauge of Green-Schwarz superstring in AdS_5 x S^5 background, we fix two bosonic variables x^{+}=\tau and y^{9}=\sigma, and then perform the partial Legendre transformation of the remaining bosonic variables. We then obtain a Lagrangian which is linear in velocity after eliminating the metric of world sheet. For such a system, one can formulate its poisson bracket and Hamiltonian. Since this system is free and without constraint, the hierarchy of infinite nonlocal conserved quantities given by Bena, Polchinski and Roiban, induce solution transformations due to Jacobi identity.