  This paper demonstrates sufficient conditions for the existence of KAM tori
in a singly thermostated, integrable hamiltonian system with $n$ degrees of
freedom with a focus on the generalized, variable-mass thermostats of order
2--which include the Nos\'e thermostat, the logistic thermostat of Tapias,
Bravetti and Sanders, and the Winkler thermostat. It extends Theorem 3.2 of
Legoll, Luskin & Moeckel, (Non-ergodicity of Nos\'e-Hoover dynamics,
Nonlinearity, 22 (2009), pp. 1673--1694) to prove that a "typical" singly
thermostated, integrable, real-analytic hamiltonian possesses a
positive-measure set of invariant tori when the thermostat is weakly coupled.
It also demonstrates a class of integrable hamiltonians, which, for a
full-measure set of couplings, satisfies the same conclusion.
