  The theory of Poisson-Lie groups and Lie bialgebras plays a major role in the
study of one dimensional integrable systems; many families of integrable
systems can be recovered from a Lax pair which is constructed from a Lie
bialgebra associated to a Poisson-Lie group. A higher homotopy notion of
Poisson-Lie groups and Lie bialgebras has been studied using Lie algebra
crossed-modules and $L_2$-algebras, which gave rise to the notion of (strict)
Lie 2-bialgebras and Poisson-Lie 2-groups . In this paper, we use these
structures to generalize the construction of a Lax pairs and introduce an
appropriate notion of {higher homotopy integrability}. Within this framework,
we introduce a higher homotopy version of the Kac-Moody algebra, with which the
2-Lax equation can be rewritten as a zero 2-curvature condition in 2+1d. An
explicit characterization of our higher Kac-Moody algebra will be given, and we
also demonstrate how it governs the 2-Lax pairs and the symmetries of a 3d
topological-holomorphic field theory studied recently. This 3d theory thus
serves as an example of a physical system that exhibits the sort of 2-graded
integrability that we have defined here.
