  We study solvable deformations of two-dimensional quantum field theories
driven by a bilinear operator constructed from a pair of conserved $U(1)$
currents $J^a$. We propose a quantum formulation of these deformations, based
on the gauging of the corresponding symmetries in a path integral. This
formalism leads to an exact dressing of the $S$-matrix of the system, similarly
as what happens in the case of a $\mathrm{T}\overline{\mathrm{T}}$ deformation.
For conformal theories the deformations under study are expected to be exactly
marginal. Still, a peculiar situation might arise when the conserved currents
$J^a$ are not well-defined local operators in the original theory. A simple
example of this kind of system is provided by rotation currents in a theory of
multiple free, massless, non-compact bosons. We verify that, somewhat
unexpectedly, such a theory is indeed still conformal after deformation and
that it coincides with a TsT transformation of the original system. We then
extend our formalism to the case in which the conserved currents are
non-Abelian and point out its connection with Deformed T-dual Models and
homogeneous Yang-Baxter deformations. In this case as well the deformation is
based on a gauging of the symmetries involved and it turns out to be
non-trivial only if the symmetry group admits a non-trivial central extension.
Finally we apply what we learned by relating the
$\mathrm{T}\overline{\mathrm{T}}$ deformation to the central extension of the
two-dimensional Poincar\'{e} algebra.
