  We construct an analytic formula of Green's function for the Laplace-Beltrami
operator on surfaces with a non-trivial Killing vector field, which contain all
surfaces of revolution. To represent the function, we explicitly build a chart
which is defined on the whole space except for at most two points, where the
metric is represented by one positive function and where the Killing vector
field is represented as a constant vector field preserving this positive
function. In this construction, we immediately obtain the formula for a given
surface with a Killing vector field by solving the geodesic equations. As
applications, we show that the logarithm of the speed and the pressure of the
Killing vector field as a steady solution of the Euler equations is equal to
the convolution of a hydrodynamic Green's function and the Gaussian curvature
and twice that up to a harmonic function. Moreover, we see that a similar
relation holds for a potential vector field on general curved surfaces. As an
example, we apply the formula to surfaces of revolution.
