Compatible Feigin–Odesskii Poisson brackets

We prove that several Feigin–Odesskii Poisson brackets associated with normal elliptic curves in $${{\mathbb {P}}}^n$$ are compatible if and only they contained a scroll or Veronese surface {P}}}^5$$ (with an exception of one case when $$n=3$$ ). In the we determine quartic corresponding to Schouten bracket two (non-compatible) $$E_1$$ $$E_2$$ .