A remark on the linearization of nondegenerate Nambu structures of coorder 1

In this note we show that Nambu structures of coorder 1 can always be linearized if they admit a closed integrable differential form. In particular, we show that a unimodular Poisson structure whose isotropy Lie algebra at a singular point is $\mathfrak{sl}(2,\mathbb{R})$, can always be linearized.