Integrable systems generated by a constant solution of the Yang-Baxter equation (Q1364873)

A class of integrable systems of the form \(q_t= r\cdot q_{xy} +F(q,q_y)\) is constructed. Here \(q(x,y,t)\) is a vector and \(r:{\mathfrak g} \to{\mathfrak g}\) is a constant linear operator on the Lie algebra \({\mathfrak g}\) that satisfies the Yang-Baxter equation \[ r \biggl(\bigl[ r(a),b\bigr] -\bigl[r(b), a\bigr]\biggr) -\bigl[r(a), r(b)\bigr] =0 \] for any \(a,b\in{\mathfrak g}\). The considered equations have local higher symmetries and conservation laws. They are similar to the \(N\)-wave equation.