Asymptotic distribution of eigenvalues for hypoelliptic systems in $R^n$

General symmetric hypoelliptic systems of differential operators in $R^n$ with discrete spectrum are considered. Two-sided estimates, as $t\to\infty$, are found for $N(t)$, the number of eigenvalues in the interval $[0,t]$. Under a regularity assumption on the behavior of the spectrum of the Weyl matrix symbol of the system, these estimates reduce to the asymptotics of $N(t)$ with an estimate of the remainder term. In part the results are also new for the scalar case.
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