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We present decomposable (5,6)-solutions $\widetilde{M}^{1,4} \times M^6$ ineleven-dimensional supergravity by solving the bosonic supergravity equationsfor a variety of non-trivial flux forms. Many of the bosonic backgroundspresented here are induced by various types of null flux forms on products ofcertain totally Ricci-isotropic Lorentzian Walker manifolds and Ricci-flatRiemannian manifolds. These constructions provide an analogue of work done byI. Chrysikos and A. Galaev who made similar computations for decomposable(6,5)-solutions. We also present bosonic backgrounds that are products ofLorentzian Einstein manifolds with negative Einstein constant (in the "mostlyplus" convention) and Riemannian K\"ahler-Einstein manifolds with positiveEinstein constant. This conclusion generalizes a result of C. N. Pope and P.van Nieuwenhuizen concerning the appearance of six-dimensionalK\"ahler-Einstein manifolds in eleven-dimensional supergravity. In this settingwe construct infinitely many non-symmetric decomposable (5, 6)-supergravitybackgrounds by using the infinitely many Lorentzian Einstein-Sasakianstructures with negative Einstein constant on the 5-sphere, known from the workof C. P. Boyer et al.

Decomposable $(5,6)$-solutions in eleven-dimensional supergravity