On a Cauchy foliation of the spacetime, Schwarzschild gravitational scattering involves well-
known integrability features based on a ‘KdV-Virasoro-Lax’ triangular structure. We study how this
structure is altered in a hyperboloidal context. In the latter, the non-selfadjoint generator of the
dynamics neatly separates between bulk and boundary contributions, which makes the possibility
of a Lax-pair formulation difficult. With integrability as a guideline, we suggest to extend the phase
space via a source term defined linearly on new degree(s) of freedom and acting on the original
ones. Following an Antonowicz-Fordy scheme, we argue that this extension leads to a Cauchy-
like situation where the corresponding spectral problem presents an isospectral multi-Hamiltonian
structure. Algebraically speaking, the resulting dynamics may be described in terms of a semi-
direct action from the bulk onto the boundary.