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Let $H(q,p) = \frac12 | p |^2 + V(q)$ be an $n$-degree of freedom $C^r$ mechanical Hamiltonian on the cotangent bundle of the $n$-torus where $r>2n+2$. When the metric $| * |$ is flat, the Nos��-thermostated system associated to $H$ is shown to have a positive-measure set of invariant tori near the infinite temperature limit. This is shown to be true for all variable mass thermostats similar to Nos��'s, too. These results complement results of Legoll, Luskin & Moeckel and the author., 11 pages

Let H(q,p) = p^2/2 + V(q) be a 1-degree of freedom mechanical Hamiltonian with a C^n periodic potential V where n>4. The Nos��-thermostated system associated to H is shown to have invariant tori near the infinite temperature limit. This is shown to be true for all thermostats similar to Nos��'s. These results complement the result of Legoll, Luskin and Moeckel who proved the existence of such tori near the decoupling limit., 10 pages