Compatible Poisson Brackets Associated with Elliptic Curves in $G(2,5)$

We prove that a pair of Feigin–Odesskii Poisson brackets on ${\mathbb P}^4$ associated with elliptic curves given as linear sections of the Grassmannian $G(2,5)$ are compatible if and only if this pair of elliptic curves is contained in a del Pezzo surface obtained as a linear section of $G(2,5)$.