A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface

Let E-G be a holomorphic principal G-bundle over a compact connected Riemann surface, where G is a connected reductive affine algebraic group defined over C, such that E-G admits a holomorphic connection. Take any beta is an element of H-0 (X, ad(E-G)), where ad(E-G) is the adjoint vector bundle for EG, such that the conjugacy class beta(x) is an element of g/G, x is an element of X, is independent of x. We give a sufficient condition for the existence of a holomorphic connection on E-G such that a is flat with respect to the induced connection on ad(E-G).