Inverse Feshbach's problem: Solvability and solutions

Given a certain specific, $M$ by $M$ matrix form of the Feshbach’s effective (i.e., energy-dependent) Hamiltonian $H_{e\text{\hspace{0.17em}⁣}f\text{\hspace{0.17em}⁣}f}^{(M)}(E)$, the inverse-problem reconstruction of the full-space, $N$ by $N$ matrix Hamiltonian $H^{(N)}$ is considered and reduced to the solution of a coupled set of polynomial algebraic equations. Using computer-assisted symbolic manipulations, an explicit algebraic reconstruction of $H^{(N)}$ is found feasible at not too large $K=N-M$.