  We construct the Lax connection of the Green-Schwarz superstring in AdS_5 x
S^5 within the Hamiltonian formalism and obtain precisely that used in
0810.4136. It differs in a crucial way from the Bena-Polchinski-Roiban
connection by terms proportional to the Hamiltonian constraints. These extra
terms ensure firstly that the integrals of motion are all first class and
secondly that the Lax connection is flat in the strong sense.
