An integrable ${\mathbb{Z}}_2^2$-graded extension of Camassa-Holm equation and its bi-Hamiltonian structure

By constructing Lax operators in the loop algebra of the ${\mathbb{Z}}_2^2$-graded extension of the Lie superalgebra $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2)$, we derive a ${\mathbb{Z}}_2^2$-graded extension of the Camassa-Holm equation. The resulting equation is an integrable nonlinear PDE for a system of four ${\mathbb{Z}}_2^2$-graded commutative functions, each associated with a distinct ${\mathbb{Z}}_2^2$-degree. We further show that the ${\mathbb{Z}}_2^2$-Camassa-Holm equation admits a bi-Hamiltonian structure. As a consequence, it possesses infinitely many conserved quantities, including one with non-trivial ${\mathbb{Z}}_2^2$-degree, which are mutually in involution with respect to the ${\mathbb{Z}}_2^2$-graded Poisson brackets.