Recall that a hypersurface $D$ in a complex manifold $X$ is called a free divisor if the Lie algebroid $\mathcal{T}_X(-\log D)$ of vector fields tangent to $D$ is locally free. This condition is equivalent to requiring that the singular locus $D_{sing}$ has codimension two in $X$ and is Cohen-Macaulay.

I am wondering what is known about an analogous situation: hypersurfaces $D \subset X$ for which $D_{sing}$ has codimension three in $X$ and is Gorenstein. Such hypersurfaces arise naturally in Poisson geometry as the degeneracy loci of generically symplectic Poisson structures. (Hence, there is a Lie algebroid present here, as well.) An example is the secant variety to an elliptic normal quintic curve $E \subset \mathbb{P}^4$, which is the degeneracy locus of one of Feigin and Odesskii's famous elliptic Poisson structures. Its singular locus is a degree-40 Gorenstein scheme supported on $E$.

I am interested in learning anything that is known about such hypersurfaces, but to single out a precise question that is relevant to Poisson geometry, I ask the following:

Which quintic hypersurfaces in $\mathbb{P}^4$ have a (possibly non-reduced) Gorenstein curve as their singular locus?

I would be happy with an answer that gives one or two examples (other than the secant variety above), and/or provides a reference to relevant literature.