We consider the linear integro-differential operator $L$ defined by
\begin{equation*}
Lu(x) =\int_{\mathbb{R}^{n}}(u(x+y)-u(x)
-\mathbbm{1}_{[1,2]}(\alpha)\mathbbm{1}_{\{|y|\leq 2\}}(y)y \cdot \nabla u(x))k(x,y)\, dy.
\end{equation*}
Here the kernel $k(x,y)$ behaves like $|y|^{-n-\alpha}$,
$\alpha \in (0,2)$,
for small $y$ and is Hölder-continuous in the first
variable, precise definitions are given below. We study the
unique solvability of the Cauchy problem corresponding to
$L$. As an application we obtain well-posedness of the martingale
problem for $L$. Our strategy follows the classical path of
Stroock-Varadhan. The assumptions allow for cases that have
not been dealt with so far.