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In this note, we prove the existence of homogeneous gradient solitons for theG$_2$-Laplacian flow by providing the first known example of this type. Thisresult singles out the G$_2$-Laplacian flow as the first known geometric flowadmitting homogeneous gradient solitons on spaces that are one-dimensionalextensions.

On the existence of homogeneous solitons of gradient type for the G$_2$-Laplacian flow