  We present the Hamiltonian formulation of the recently constructed integrable
theories of arXiv:2006.12525. These theories turn out to be canonically
equivalent to the sum of an asymmetrically gauged CFT and of the most general
$\lambda$-deformed model of arXiv:1812.04033. Using the Hamiltonian formalism,
we prove that the full set of conserved charges of the models of
arXiv:2006.12525 are in involution, ensuring their Hamiltonian integrability.
Finally, we show that the equations of motion of these theories can be put in
the form of zero curvature Lax connections.
