  We introduce the class of variable anisotropic singular integral operators
associated to a continuous multi-level ellipsoid cover $\Theta$ of
$\mathbb{R}^n$ introduced by Dahmen, Dekel, and Petrushev \cite{ddp}. This is
an extension of the classical isotropic singular integral operators on
$\mathbb{R}^n$ of arbitrary smoothness and their anisotropic analogues for
general expansive matrices introduced by the first author \cite{b}. We
establish the boundedness of variable anisotropic singular integral operators
$T$ on the Hardy spaces with pointwise variable anisotropy $H^p(\Theta)$, which
were developed by Dekel, Petrushev, and Weissblat \cite{dpw}. In contrast with
the general theory of Hardy spaces on spaces of homogenous type, our results
work in the full range $0<p\leq 1$.
