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

Classification of Parameter-Dependent Quantum Integrable Models, Their Parameterization, Exact Solution, and Other Properties