Abstract
This work investigates the integrability of an extended (3+1)-dimensional hierarchy of nonlinear evolution equations (NLEEs), which include the sixth-order Korteweg-de Vries (KdV6) equation. These NLEEs describe several nonlinear phenomena in fluid dynamics and in other physical settings. The integrability of these equations is demonstrated via applying the Painleve analysis. We show that these equations are characterized by distinct dispersion relations. By applying Hirota's direct method, several soliton solutions for each equation are obtained and for the general case too.