A Reviewer made the following comment regarding the term "proper degeneracy"
<review>I think this is just the notion of a degenerate system, but not necessarily of a properly degenerate one. In my understanding of Arnold's paper, proper degeneracy means that a degenerate system becomes nondegenerate by incorporating the average of the perturbation to first order and pushing the remaining perturbation to second order. </review>
However, the common usage of "proper degeneracy" is just that of an integrable system having the Hamiltonian in action variables, whose hessian matrix has an identically vanishing determinant; compare, e.g., [V.I. Arnold, V.V. Kozlov and A.I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Third Edition, Springer 2006, Sec. 5.2.1, p. 181