<p>In this paper, we consider two indices that are independently applied in the study of vector fields on analytic singular surfaces, namely the Conley index and the Schwartz index. While the first requires that the invariant sets of the vector field be isolated, the second requires the vector field to be radial. We focus on a class of continuous vector fields on the two-dimensional case, where both indices can be computed simultaneously, to analyze how each index successfully measures the Euler characteristic of the singular variety, despite yielding different local results. Additionally, we present a new topological formula that relates the Betti numbers of the links of the singularities in an analytic singular surface.</p>