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We study existence of densities for solutions to stochastic differentialequations with H\"older continuous coefficients and driven by a $d$-dimensionalL\'evy process $Z=(Z_{t})_{t\geq 0}$, where, for $t>0$, the density function$f_{t}$ of $Z_{t}$ exists and satisfies, for some$(\alpha_{i})_{i=1,\dots,d}\subset(0,2)$ and $C>0$, \begin{align*}  \limsup\limits _{t \to 0}t^{1/\alpha_{i}}\int\limits_{\mathbb{R}^{d}}|f_{t}(z+e_{i}h)-f_{t}(z)|dz\leq C|h|,\ \ h\in \mathbb{R},\ \i=1,\dots,d. \end{align*} Here $e_{1},\dots,e_{d}$ denote the canonical basisvectors in $\mathbb{R}^{d}$. The latter condition covers anisotropic$(\alpha_{1},\dots,\alpha_{d})$-stable laws but also particular cases ofsubordinate Brownian motion. To prove our result we use some ideas taken from\citep{DF13}.

Existence of densities for stochastic differential equations driven by L\'evy processes with anisotropic jumps