  The objective of this work is to explore the class of equations of the
Non-linear Schrodinger type by employing the Adler-Kostant-Symes theorem and
the Tu methodology.In the first part of the work, the AKS theory is discussed
in detail showing how to obtain the non-linear equations starting from a
suitably chosen spectral problem.Equations derived by this method include
different members of the NLS family like the NLS, the coupled KdV type NLS, the
generalized NLS, the vector NLS, the Derivative NLS, the Chen-Lee-Liu and the
Kundu-Eckhaus equations. In the second part of the paper, the steps in the Tu
methodology that are used to formulate the hierarchy of non-linear evolution
equations starting from a spectral problem, are outlined. The AKNS,
Kaup-Newell, and generalized DNLS hierarchies are obtained by using this
algorithm. Several reductions of the hierarchies are illustrated. The famous
trace identity is then applied to obtain the Hamiltonian structure of these
hierarchies and establish their complete integrability. In the last part of the
paper, the non-holonomic deformation of the class of integrable systems
belonging to the NLS family is studied. Equations examined include the NLS,
coupled KdV-type NLS and Derivative NLS (both Kaup-Newell and Chen-Lee-Liu
equations). NHD is also applied to the hierarchy of equations in the AKNS
system and the KN system obtained through application of the Tu
methodology.Finally, we discuss the connection between the two formalisms and
indicate the directions of our future endeavour in this area.
