  We derive reconstruction formulas for a family of geodesic ray transforms
with connection, defined on simple Riemannian surfaces. Such formulas provide
injectivity of such all transforms in a neighbourhood of constant curvature
metrics and non-unitary connections with curvature close to zero. If certain
Fredholm equations are injective in the absence of connection, then for any
smooth enough connection multiplied by a complex parameter, the corresponding
transform is injective for all values of that parameter outside a discrete set.
Range characterizations are also provided, as well as numerical illustrations.
