Two-Component Wadati--Konno--Ichikawa Equation and Its Symmetry Reductions

It is shown that two-component Wadati--Konno--Ichikawa (WKI) equation, i.e.~a generalization of the well-known WKI equation, is obtained from the motion of space curves in Euclidean geometry, and it is exactly a system for the graph of the curves when the curve motion is governed by the two-component modified Korteweg--de Vries flow. Group-invariant solutions of the two-component WKI equation which corresponds to an optimal system of its Lie point symmetry groups are obtained, and its similarity reductions to systems of ordinary differential equations are also given.