  The conformal algebra in 2D (Diff($S^{1}$)$\oplus$Diff($S^{1}$)) is shown to
be preserved under a nonlinear map that mixes both chiral (holomorphic)
generators $T$ and $\bar{T}$. It depends on a single real parameter and it can
be regarded as a ``nonlinear $SO(1,1)$ automorphism.'' The map preserves the
form of the momentum density and naturally induces a flow on the energy density
by a marginal $\sqrt{T\bar{T}}$ deformation. In turn, the general solution of
the corresponding flow equation of the deformed action can be analytically
solved in closed form, recovering the nonlinear automorphism. The deformed
CFT$_{2}$ can also be described through the original theory on a
field-dependent curved metric whose lapse and shift functions are given by the
variation of the deformed Hamiltonian with respect to the energy and momentum
densities, respectively. The conformal symmetries of the deformed theories can
then also be seen to arise from diffeomorphisms that fulfill suitably deformed
conformal Killing equations. Besides, Cardy formula is shown to map to itseft
under the nonlinear automorphism. As a simple example, the deformation of $N$
free bosons is briefly addressed, making contact with recent related results
and the dimensional reduction of the ModMax theory. Furthermore, the nonlinear
map between the conformal algebra in 2D and its ultra/non-relativistic versions
(BMS$_{3}$$\approx$CCA$_{2}$$\approx$GCA$_{2}$), including the corresponding
finite $\sqrt{T\bar{T}}$ deformation, are recovered from a limiting case of the
nonlinear automorphism. The extension to a three-parameter nonlinear $ISO(1,1)$
automorphism of the conformal algebra, and a discrete nonlinear automorphism of
BMS$_{3}$ are also briefly discussed.
