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IN-DNLS considered here is a countable infinite set of coupled one-dimensional nonlinear ordinary differential difference equations with a tunable nonintegrability parameter. When this parameter vanishes, IN-DNLS reduces to the famous integrable Ablowitz-Ladik (AL) equation. The formation of unstaggered and staggered stationary  localized states (SLSs) in IN-DNLS is studied here using a discrete variational method. The functional form of  stationary soliton of AL equation is used as the ansatzfor SLSs. Derivation of the appropriate functional and its equivalence to the effective Lagrangian are presented. Formation of on-site peaked and  intersite peaked unstaggered SLSs and their dependence on the nonintegrability parameter are investigated. On-site peaked states are found to be energetically stable. Results are explained using the effective mass picture. Also, the properties of staggered SLSs of Sievers-Takeno- (ST-) likemode and Page- (P-) like mode are investigated and explained using the same effective mass picture. It is further shown here that an unstable SLS which is found in the truncated analysis of the problem does not survive in the exact calculation. For large-width and small-amplitude SLSs, the known asymptotic result for the amplitude is obtained. Further scope and possible extensions of this work are discussed

A discrete variational approach for investigation of stationary localized states in a discrete nonlinear Schrödinger equation, named IN-DNLS