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]
Nearby in the stack
on the punctual Hilbert scheme of points
X[n]
has an induced structure of a
((V∨)[n])∨
-twisted Hitchin pair, where
(V∨)[n]
is a vector bundle on
X[n]
constructed using the dual
V∨
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.