Infinite pseudo-conformal symmetries of classical $T \bar T$, $J \bar T $ and $J T_a$ - deformed CFTs

We show that $T \bar T, J T$ and $J T_a$ - deformed classical CFTs possess an infinite set of symmetries take the form a field-dependent generalization two-dimensional conformal transformations. If, in addition, seed affine $U(1)$ symmetry, we it also survives theories, again form. These can be understood as infinitely-extended underlying CFT, seen through prism "dynamical coordinates" characterise each these deformations. compute Poisson bracket algebra associated conserved charges, using Hamiltonian formalism. In case deformations, find two copies functional Witt Kac-Moody algebra. deformation, is possible to obtain commuting