$N=2$ supersymmetric $a=4$-Korteweg–de Vries hierarchy derived via Gardner’s deformation of Kaup–Boussinesq equation

We consider the problem of constructing Gardner’s deformations for the $N=2$ supersymmetric $a=4$-Korteweg–de Vries (SKdV) equation; such deformations yield recurrence relations between the super-Hamiltonians of the hierarchy. We prove the nonexistence of supersymmetry-invariant deformations that retract to Gardner’s formulas for the Korteweg-de Vries (KdV) with equation under the component reduction. At the same time, we propose a two-step scheme for the recursive production of the integrals of motion for the $N=2$⁠, $a=4$-SKdV. First, we find a new Gardner’s deformation of the Kaup–Boussinesq equation, which is contained in the bosonic limit of the superhierarchy. This yields the recurrence relation between the Hamiltonians of the limit, whence we determine the bosonic super-Hamiltonians of the full $N=2$⁠, $a=4$-SKdV hierarchy. Our method is applicable toward the solution of Gardner’s deformation problems for other supersymmetric KdV-type systems.