The method of <i>M_n</i>-extension: The KdV equation

In this work we generalize M-2-extension that has been introduced recently. For illustration we use the KdV equation. We present five different M-3-extensions of the KdV equation and their recursion operators. We give a compact form of M-n-extension of the KdV equation and recursion operator of the coupled KdV system. The method of M-n-extension can be applied to any integrable scalar equation to obtain integrable multi-field system of equations. We also present unshifted and shifted nonlocal reductions of an example of M-3-extension of KdV.