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The Hamiltonian methods of the theory of solitons are applied togravisolitons in the dressing formalism. The Poisson bracket for theLie-algebra valued one-form $A(\varsigma, \eta,\gamma)=\Psi_{,\gamma}\Psi^{-1}$, for gravisolitons in the dressing formalism,for a specific background solution, is defined and computed, agreeing withresults previously obtained. A transition matrix ${\cal T}=A(\varsigma, \eta,\gamma) A^{-1}(\eta, \xi, -\gamma)$ for $A$ is defined relating $A$ at ingoingand outgoing light cones. It is proved that it satisfies equations familiarfrom integrable pde's with the role of time played by the null coordinate$\eta$. This is a new result mathematically, since there has not been atransition matrix for $A$ in the litterature, while physically it presents thepossibility of obtaining integrals of motion (for appropriate boundaryconditions), from the trace of the derivative with respect to the nullcoordinate $\eta$, of ${\cal T}$, in terms of classical relativity connections,since $A(\gamma = \pm 1)$ can be expressed in a simple way in terms of theclassical Christoffel symbols. This may prove of use upon quantization sinceconnections are fundamental variables of quantum gravity. The roles of $\eta$and $\varsigma $ may be reversed to obtain integrals of motion for $\varsigma$,thus $\varsigma$ playing the role of time. This ties well with the two-timeinterpretation and approach already established before.

Transition Matrix, Poisson Bracket for gravitational solitons in the dressing formalism