  This paper is devoted to the study of flows associated to non-smooth vector
fields. We prove the well-posedness of regular Lagrangian flows associated to
vector fields $\mathbf{B}=(\mathbf{B}^1,...,\mathbf{B}^d)\in
L^1(\mathbb{R}_+;L^1(\mathbb{R}^d)+L^\infty(\mathbb{R}^d))$ satisfying $
\mathbf{B}^i=\sum_{j=1}^{m}\mathbf{K}_j^i*b_j,$ $b_j\in
L^1(\mathbb{R}_+,BV(\mathbb{R}^d))$ and $\operatorname{div}(\mathbf{B})\in
L^1(\mathbb{R}_+;L^\infty(\mathbb{R}^d))$ for $d,m\geq 2$, where
$(\mathbf{K}_j^i)_{i,j}$ are singular kernels in $\mathbb{R}^d$. Moreover, we
also show that there exist an autonomous vector-field $\mathbf{B}\in
L^1(\mathbb{R}^2)+L^\infty(\mathbb{R}^2)$ and singular kernels
$(\mathbf{K}_j^i)_{i,j}$, singular Radon measures $\mu_{ijk}$ in $\mathbb{R}^2$
satisfying $\partial_{x_k}
\mathbf{B}^i=\sum_{j=1}^{m}\mathbf{K}_j^i\star\mu_{ijk}$ in distributional
sense for some $m\geq 2$ and for $k,i=1,2$ such that regular Lagrangian flows
associated to vector field $\mathbf{B}$ are not unique.
