$N=4$ superextension of the Liouville equation with quaternion structure

An $N=4$ supersymmetric extension of the Liouville equation is constructed. It has internal $SU(2)\times SU(2)$ gauge symmetry and can be adequately formulated in terms of a real quaternion $N=4$ superfield on which definite conditions of Grassmann analytieity are imposed. Both the dynamical equations as well as the analyticity conditions follow from the zero-curvature representation on the superalgebra $su(1,1|2)$. It is shown that the obtained system is invariant with respect to transformations of the infinite-dimensional superalgebra of the $SU(2)$ superstring, the realization of these transformations differing from those previously known. The possible connection between the $N=4$ Liouville equation and the theory of the $SU(2)$ superstring is discussed.