A criterion for holomorphic Lie algebroid connections

Given a holomorphic Lie algebroid $(V,\phi )$ on a compact connected Riemann surface $X$, we give a necessary and sufficient condition for a holomorphic vector bundle $E$ on $X$ to admit a holomorphic Lie algebroid connection. If $(V,\phi )$ is nonsplit, then every holomorphic vector bundle on $X$ admits a holomorphic Lie algebroid connection for $(V,\phi )$. If $(V,\phi )$ is split, then a holomorphic vector bundle $E$ on $X$ admits a holomorphic Lie algebroid connection if and only if the degree of each indecomposable component of $E$ is zero.