Painlev, analysis for a new integrable equation combining the modified Calogero-Bogoyavlenskii-Schiff (MCBS) equation with its negative-order form

A new integrable equation is constructed by combining the recursion operator of the modified Calogero-Bogoyavlenskii-Schiff equation and its inverse recursion operator. The Painlev, is performed to demonstrate the complete integrability of the newly developed equation. Multiple-soliton solutions are depicted as manifestation of the integrability. We further show that this equation enjoys a variety of soliton solutions that include kinks, peakon, cuspon.