  We prove the modified algebraic Bethe Ansatz characterization of the spectral
problem for the closed XXX Heisenberg spin chain with an arbitrary twist and
arbitrary positive (half)-integer spin at each site of the chain. We provide
two basis to characterize the spectral problem and two families of
inhomogeneous Baxter T-Q equations. The two families satisfy an inhomogeneous
quantum Wronskian equation.
