  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.
