Integrable $\mathbb{Z}_2^2$-graded super-Liouville Equation and Induced $\mathbb{Z}_2^2$-graded super-Virasoro Algebra

We present a framework for enlarging the construction of $\mathbb{Z}_2^2$-graded classical Toda theory from the class of $\mathbb{Z}_2^2$-graded Lie algebras to the class of $\mathbb{Z}_2^2$-graded Lie superalgebras. This scheme is applied to derive a $\mathbb{Z}_2^2$-graded extension of the super-Liouville equation based on a $\mathbb{Z}_2^2$-graded extension of $\mathfrak{osp}(1|2).$ The mathematical tools employed in this work are a $\mathbb{Z}_2^2$-graded version of the zero-curvature formalism and of the Polyakov's soldering procedure. It is demonstrated that both methods yield the same $\mathbb{Z}_2^2$-graded super-Liouville equation. An algebraic construction of solutions to the resulting equations is also presented, together with their B\"acklund transformations. Furthermore, three distinct new $\mathbb{Z}_2^2$-graded extensions of the super-Virasoro algebra are obtained via Hamiltonian reduction of the WZNW currents defined for $\mathbb{Z}_2^2$-$\mathfrak{osp}(1|2).$