We introduce the class of variable anisotropic singular integral operators associated to a continuous multi-level ellipsoid cover $$\Theta $$ $$\mathbb {R}^n$$ introduced by Dahmen et al. (Constr Approx 31:149–194, 2010). This is an extension classical isotropic on $${\mathbb R}^n$$ arbitrary smoothness and their analogues for general expansive matrices first author Bownik (Mem Am Math Soc 164(...