New exotic solitary wave and the painlevé integrability of one type of the nonlinear dispersive generalized DGH equation are studied. By Painlevé analysis, we discover that the nonlinear dispersive generalized DGH equation with m＝2 is integrable, which is a new integrable equation. By the new variable transformation and the auto-Backlund transformation，we obtain abundant exotic solitary wave solutions, such as compactons, peakons, new double solitary waves with peak points, and double solitary waves with blow-up points.