  We introduce and study a novel class of classical integrable many-body
systems obtained by generalized $T\bar{T}$-deformations of free particles.
Deformation terms are bilinears in densities and currents for the continuum of
charges counting asymptotic particles of different momenta. In these models,
which we dub ``semiclassical Bethe systems'' for their link with the dynamics
of Bethe ansatz wave packets, many-body scattering processes are factorised,
and two-body scattering shifts can be set to an almost arbitrary function of
momenta. The dynamics is local but inherently different from that of known
classical integrable systems. At short scales, the geometry of the deformation
is dynamically resolved: either particles are slowed down (more space
available), or accelerated via a novel classical particle-pair
creation/annihilation process (less space available). The thermodynamics both
at finite and infinite volumes is described by the equations of (or akin to)
the thermodynamic Bethe ansatz, and at large scales generalized hydrodynamics
emerge.
