  We study general quantum integrable Hamiltonians linear in a coupling
constant and represented by finite NxN real symmetric matrices. The restriction
on the coupling dependence leads to a natural notion of nontrivial integrals of
motion and classification of integrable families into Types according to the
number of such integrals. A Type M family in our definition is formed by N-M
nontrivial mutually commuting operators linear in the coupling. Working from
this definition alone, we parameterize Type M operators, i.e. resolve the
commutation relations, and obtain an exact solution for their eigenvalues and
eigenvectors. We show that our parameterization covers all Type 1, 2, and 3
integrable models and discuss the extent to which it is complete for other
types. We also present robust numerical observation on the number of energy
level crossings in Type M integrable systems and analyze the taxonomy of types
in the 1d Hubbard model.
