Connections of the coupled two-component Harry Dym system with the coupled two-component KdV equation

Four different two-component coupled Harry Dym equations and their Lax representations are given. All these equations are transformed, via the reciprocal link, to the integrable coupled KdV equation. The first two-component Harry Dym equation is connected with the Hirota–Satsuma equation, the second one is connected with the Drinfeld–Sokolov equation, the third one gives us a system of coupled KdV equations considered by Drinfeld and Sokolov, while the last one reduces to the know symmetric system of coupled KdV equations.