  We investigate the connection between energy level crossings in integrable
systems and their integrability, i.e. the existence of a set of non-trivial
integrals of motion. In particular, we consider a general quantum Hamiltonian
linear in the coupling u, H(u) = T + uV, and require that it has the maximum
possible number of nontrivial commuting partners also linear in u. We
demonstrate how this commutation requirement alone leads to: (1) an exact
solution for the energy spectrum and (2) level crossings, which are always
present in these Hamiltonians in violation of the Wigner-von Neumann
non-crossing rule. Moreover, we construct these Hamiltonians explicitly by
resolving the above commutation requirement and show their equivalence to a
sector of Gaudin magnets (central spin Hamiltonians). In contrast, fewer than
the maximum number of conservation laws does not guarantee level crossings.
