In this paper, we study the following Levi-Civitá equation $$ w(xy) + w(yx) = \sum_{i=1}^{m} f_{i}(x)g_{i}(y) \quad \quad \quad ({\rm LC})$$ on a compact group G, where w, f i ’s, and g i ’s are continuous complex-valued functions to determine. Our main ingredient is (nonabelian) Fourier analysis on compact groups. We apply the Fourier transform to Eq. (LC) on the product group G × G so that we obtain its several equivalent operator equations. Using those equivalent equations, we derive some crucial properties of solutions to Eq. (LC). Consequently, Eq. (LC) with m ≤ 2 is completely solved. In particular, a Wilson type equation arising from and playing a central role in [4] is solved on compact groups.