Two-fold integrable hierarchy of nonholonomic deformation of the DNLS and the Lenells-Fokas equation

The concept of the nonholonomic deformation formulated recently for the AKNS family is extended to the Kaup-Newell class. Applying this construction we discover a novel two-fold integrable hierarchy related to the deformed derivative nonlinear Schr\"odinger (DNLS) equation and found the exact soliton solutions exhibiting unusual accelerating motion for both its field and the perturbing functions. Extending the idea of deformation the integrable perturbation of the gauge related Chen-Lee-Liu DNLS equation is constructed together with its soliton solution. We show that, the recently proposed Lenells-Fokas (LF) equation falls in the deformed DNLS hierarchy, sharing the accelerating soliton and other unusual features. Higher order integrable deformations of the LF and the DNLS equations are proposed.