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. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 305–309, February, 2000.