A direct method to construct integrals for an Nth-order autonomous ordinary difference equation (ODE ): w nCN ZF(w n , ., w nCNK1 ) is presented. As an illustration we first consider third-order autonomous ODE w nC3 ZF(w n , w nC1 , w nC2 ) and identify the forms of F for which two independent integrals exist. The effectiveness of the method to construct two or more integrals for fourth-and fifth-order ODEs is also demonstrated. The question of integrability of each of the identified difference equations with more than one integral is also discussed.

A direct method to construct polynomial integrals for third order ordinary difference equation (O∆E) w(n + 3) = F (w(n), w(n + 1), w(n + 2)) and fourth order O∆E w(n + 4) = F (w(n), w(n + 1), w(n + 2), w(n + 3)) is presented. The effectiveness of the method to construct more than one polynomial integral for N-th order O∆E is also briefly discussed.

In this paper, a direct method is proposed to construct conservation laws for nonautonomous nonlinear partial difference equations (PΔEs). The identified PΔEs can be classified into QRT type, KdV type, mKdV type and sG type. The integrability of the identified partial difference equations is also discussed.

A systematic investigation to derive three-dimensional analogs of two-dimensional Quispel, Roberts and Thompson (QRT) mappings is presented. The question of integrability of the obtained three-dimensional mappings with two independent integrals is also analyzed. It is also shown that there exist three-dimensional QRT maps with three n-dependent integrals.

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In this paper an attempt is made to find four-dimensional analogs of two-dimensional Quispel, Roberts and Thompson mappings and identified four distinct cases have been identified. The obtained mappings are measure preserving. The integrability of the isolated mappings is examined by constructing a sufficient number of integrals and their symplectic structure wherever possible.

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