  The Green's function approach of Giles and Pierce is used to build the lift
and drag based analytic adjoint solutions for the two-dimensional
incompressible Euler equations around irrotational base flows. The drag-based
adjoint solution turns out to have a very simple closed form in terms of the
flow variables and is smooth throughout the flow domain, while the lift-based
solution is singular at rear stagnation points and sharp trailing edges owing
to the Kutta condition. This singularity is propagated to the whole dividing
streamline (which includes the incoming stagnation streamline and the wall)
upstream of the rear singularity (trailing edge or rear stagnation point) by
the sensitivity of the Kutta condition to changes in the stagnation pressure.
