Generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>T</mml:mi><mml:mover accent="true"><mml:mi>T</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover></mml:math>-like flows for scalar theories in two dimens

We demonstrate that the necessary condition for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>S</a:mi><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:mi>N</a:mi><a:mo stretchy="false">)</a:mo><a:mo>×</a:mo><a:mi>S</a:mi><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:mi>N</a:mi><a:mo stretchy="false">)</a:mo></a:math> duality invariance manifests as a partial differential equation in two-dimensional scalar theories. This condition, expressed as a partial differential equation, corresponds precisely to the integrability condition. We derive a general perturbation solution to this partial differential equation, which includes both a root <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>T</g:mi><g:mover accent="true"><g:mi>T</g:mi><g:mo stretchy="false">¯</g:mo></g:mover></g:math> flow equation and an irrelevant <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>T</k:mi><k:mover accent="true"><k:mi>T</k:mi><k:mo stretchy="false">¯</k:mo></k:mover></k:math>-like flow equation. Additionally, we identify a general form for these flow equations that commute with each other. Our results establish a general integrable theory characterized by theory-dependent coefficients at each order in the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>λ</o:mi></o:math> expansion. This unified framework systematically classifies all integrable theories possessing two Lorentz-invariant variables (<q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>P</q:mi><q:mn>1</q:mn></q:msub></q:math>, <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>P</s:mi><s:mn>2</s:mn></s:msub></s:math>) while accommodating arbitrary orders of the coupling constants (<u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>λ</u:mi></u:math>, <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>γ</w:mi></w:math>). The theory provides a comprehensive classification scheme that encompasses both known and novel integrable systems within this class.