Symmetries of Systems of the Hyperbolic Riccati Type

Let $\mathfrak G=\bigoplus_{i\in\mathbb Z}\mathfrak G_i$ be a Kac–Moody algebra, $U(x,y)$ be a function defined in $\mathfrak G_{-1}$, and $a$ be a constant element of $\mathfrak G_1$. We prove that the equation $U_{xy}=\bigl[[U,a],U_x\bigr]$ has two symmetry hierarchies connected by a gauge transformation. In particular, the well-known Konno equation appears in the case of the algebra $A_1^{(1)}$. The corresponding symmetry hierarchies contain the nonlinear Schrödinger and the Heisenberg magnet equations.