Supersymmetric ${\alpha }^{\mathrm{\prime }}$-corrections to the Generalized Kerr-Schild Ansatz

We investigate the supersymmetric extension of the generalized Kerr-Schild ansatz (gKSA) in Double Field Theory (DFT) including first-order ${\alpha }^{\mathrm{\prime }}$ corrections. Supersymmetry plays a central role in constraining higher-derivative deformations, and in this work we focus on the structure of the $\mathcal{O}({\alpha }^{\mathrm{\prime }})$ Killing Spinor Equations (KSEs). Starting from the ${\alpha }^{\mathrm{\prime }}$-corrected supersymmetry transformations of the fermionic fields, we derive the corresponding KSEs in DFT and analyze their behavior under the gKSA. The generalized Green-Schwarz transformations impose specific constraints on the perturbations that ensure the linearization of the ${\alpha }^{\mathrm{\prime }}$-corrected KSEs. We then provide the explicit parameterization of the results in terms of the ten-dimensional $\mathcal{N}=1$ supergravity fields. Our construction establishes a unified framework for exploring supersymmetric backgrounds in ${\alpha }^{\mathrm{\prime }}$-corrected DFT and offers new tools for generating higher-derivative supergravity solutions.