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Surprising links between the deformation of 2D quantum field theories inducedby the composite $\textrm{T} \bar{\textrm{T}}$ operator, effective stringmodels and the $AdS/$CFT correspondence, have recently emerged. The purpose ofthis article is to discuss various classical aspects related to the deformationof 2D interacting field theories. Special attention is given to thesin(h)-Gordon model, for which we were able to construct the $\textrm{T}\bar{\textrm{T}}$-deformed Lax pair. We consider the Lax pair formulation to bethe first essential step toward a more satisfactory geometrical interpretationof this deformation within the integrable model framework.  Furthermore, it is shown that the 4D Maxwell-Born-Infeld theory, possiblywith the addition of a mass term or a derivative-independent potential,corresponds to a natural extension of the 2D examples. Finally, we brieflycomment on 2D Yang-Mills theory and propose a modification of the heat kernel,for a generic surface with genus $p$ and $n$ boundaries, which fully accountsfor the $\textrm{T} \bar{\textrm{T}}$ contribution.

Generalised Born-Infeld models, Lax operators and the $\textrm{T} \bar{\textrm{T}}$ perturbation