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Abstract Nonlocal integrable partial differential equations possessing a spatial or temporal reflection have constituted an active research area for the past decade. Recently, more general classes of these nonlocal equations have been proposed, wherein the nonlocality appears as a combination of a shift (by a real or a complex parameter) and a reflection. This new shifting parameter manifests itself in the inverse scattering transform (IST) as an additional phase factor in an analogous way to the classical Fourier transform. In this paper, the IST is analyzed in detail for several examples of such systems. Particularly, time, space, and space‐time‐shifted nonlinear Schrödinger (NLS) and space‐time‐shifted modified Korteweg‐de Vries equations are studied. Additionally, the semidiscrete IST is developed for the time, space, and space‐time‐shifted variants of the Ablowitz–Ladik integrable discretization of the NLS. One‐soliton solutions are constructed for all continuous and discrete cases. 

A fractional extension of the integrable Toda lattice with decaying data on the line is obtained. Completeness of squared eigenfunctions of a linear discrete real tridiagonal eigenvalue problem is derived. This completeness relation allows nonlinear evolution equations expressed in terms of operators to be written in terms of underlying squared eigenfunctions and is related to a discretization of the continuous Schrödinger equation. The methods are discrete counterparts of continuous ones recently used to find fractional integrable extensions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger (NLS) equations. Inverse scattering transform (IST) methods are used to linearize and find explicit soliton solutions to the integrable fractional Toda (fToda) lattice equation. The methodology can also be used to find and solve fractional extensions of a Toda lattice hierarchy.