  We present a general approach to the parametrix construction. We apply it to
prove the uniqueness and existence of a weak fundamental solution for the
equation $\partial_t =\mathcal{L}$ with non-symmetric non-local operators $$
\mathcal{L}f(x):= b(x)\cdot \nabla f(x)+ \int_{\mathbb{R}^d}( f(x+z)-f(x)-
1_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z)\, dz\,, $$ under certain
assumptions on $b$, $\kappa$ and $J$. The result allows more general
coefficients even for $J(z)=|z|^{-d-1}$.
