Classification of integrable \(\mathcal B\)-equations (Q1881150)

The author gives a classification of integrable \(\mathcal B\)-equations of coupled evolution equations of the form \(u_t=a_1u_n+K(v_0,v_1,\ldots)\), \(v_t=a_2v_n\), where \(a_1\) and \(a_2\) are constant and \(K\) is a quadratic polynomial in derivatives of \(v\), having an infinite number of generalized symmetries. The existence of a certain finite number of integrable \(\mathcal B\)-equations at every order is proved. The question of whether the hierarchies of symmetries are exhaustive is also tackled; furthermore, the author shows how to determine whether a \(\mathcal B\)-equation is a member of the hierarchy of another equation, and gives formulae for the number of new integrable equations at arbitrary order.