In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator ${\mathcal L} \colon Fw \rightarrow Fw$ where $F$ is the ring of symmetric functions and $w$ is a variable. In this paper, we (1) establish a cyclic decomposition $Fw \cong \bigoplus_{\lambda} Z(j_{\lambda}, {\mathcal L})$ into finite-dimensional ${\mathcal L}$-cyclic subspaces in which Jack polynomials $j_{\lambda}$ may be taken as cyclic vectors and (2) prove that the restriction of ${\mathcal L}$ to each $Z(j_{\lambda}, {\mathcal L})$ has simple spectrum given by the anisotropic contents $s$ of the addable corners $s$ of the Young diagram of $\lambda$. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated to ${\mathcal L}$, both established by Nazarov-Sklyanin. Finally, we conjecture that the ${\mathcal L}$-eigenfunctions $\psi_{\lambda}^s {\in Fw}$ {with eigenvalue $s$ and constant term} $\psi_{\lambda}^s|_{w=0} = j_{\lambda}$ are polynomials in the rescaled power sum basis $V_{\mu} w^l$ of $Fw$ with integer coefficients.