  Given a vector bundle $E$ on a smooth projective curve or surface $X$
carrying the structure of a $V$-twisted Hitchin pair for some vector bundle
$V$, we observe that the associated tautological bundle $E^{[n]}$ on the
punctual Hilbert scheme of points $X^{[n]}$ has an induced structure of a
$((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a
vector bundle on $X^{[n]}$ constructed using the dual $V^\vee$ of $V$. In
particular, a Higgs bundle on $X$ induces a logarithmic Higgs bundle on the
Hilbert scheme $X^{[n]}$. We then show that the known results on stability of
tautological bundles and reconstruction from tautological bundles generalize to
tautological Hitchin pairs.
