Lax representations via twisted extensions of infinite-dimensional Lie algebras: some new results

Abstract: We apply the technique of twisted extensions of infinite-dimensional Lie algebras to find new 3D integrable {\sc pde}s related to the deformations of Lie algebra $\mathbb{R}_N[s]\otimes \mathfrak{w}$ with $N=1, 2$ as well as to the Lie algebra $\mathfrak{h} \oplus \mathfrak{w}$, where $\mathbb{R}_N[s]$ is the algebra of truncated polynomials of degree $N$, $\mathfrak{w}$ is the Lie algebra of polynomial vector fields on $\mathbb{R}$ and $\mathfrak{h}$ is the Lie algebra of polynomial Hamiltonian vector fields on $\mathbb{R}^2$.