| Summary: | We consider the possibility of constructing a hierarchy of the complex extension of the Korteweg–de Vries equation (cKdV), which under the assumption that the function is real passes into the KdV hierarchy. A hierarchy is understood here as a family of nonlinear partial differential equations with a Lax pair with a common scattering operator. The cKdV hierarchy is obtained by examining the equation on the eigenvalues of the fourth-order Hermitian self-conjugate operator on the invariant transformations of the eigenvector-functions. It is proved that for an operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>H</mi><mo stretchy="false">^</mo></mover><mi>n</mi></msub></mrow></semantics></math></inline-formula> to transform a solution of the equation on eigenvalues <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced><mrow><mover accent="true"><mi>M</mi><mo stretchy="false">^</mo></mover><mo>−</mo><mi>λ</mi><mi>E</mi></mrow></mfenced><mi>V</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> into a solution of the same equation, it is necessary and sufficient that the complex function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow></semantics></math></inline-formula> of the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>M</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> satisfies special conditions that are the complexifications of the KdV hierarchy equations. The operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>H</mi><mo stretchy="false">^</mo></mover><mi>n</mi></msub></mrow></semantics></math></inline-formula> are constructed as differential operators of order 2<i>n</i> + 1. We also construct a hierarchy of perturbed KdV equations (pKdV) with a special perturbation function, the dynamics of which is described by a linear equation. It is based on the system of operator equations obtained by Bogoyavlensky. Since the elements of the hierarchies are united by a common scattering operator, it remains unchanged in the derivation of the equations. The second differential operator of the Lax pair has increasing odd derivatives while retaining a skew-symmetric form. It is shown that when perturbation tends to zero, all hierarchy equations are converted to higher KdV equations. It is proved that the pKdV hierarchy equations are a necessary and sufficient condition for the solutions of the equation on eigenvalues to have invariant transformations.
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