Integrable super extensions of $K(-2,-2)$ equation

Two coupled systems involving both bosonic and fermionic fields are proposed as super generalizations of the $K(-2,-2)$ equation $u_t=\partial_x^3(u^{-2}/2)-\partial_x(2u^{-2})$. Linear spectral problems are presented to certify their integrability and lead to infinitely many conservation laws. Based on natural conservation laws, reciprocal transformations are defined that map one super $K(-2,-2)$ equation to Kupershmidt's super modified Korteweg–de Vries (mKdV) equation, and the other super $K(-2,-2)$ equation to the supersymmetric mKdV equation. By means of these connections, bi-Hamiltonian formulations are established for the super $K(-2,-2)$ equations.