A quest for the integrable equation in $3+1$ dimensions

The $L$ and $T$ operators of the Korteweg–de Vries equation are modified to seek a $(3+1)$-dimensional integrable equation. However, the Lax equation in this case is eventually reduced to a $(2+1)$-dimensional equation. We also propose other modified equations and their Lax pairs. A similar attempt is made to derive a higher-dimensional Harry Dym (HD) equation. As a result, a new $(2+1)$-dimensional HD equation is presented.