  We present a study of complex singularities of a two-parameter family of
solutions for the two-dimensional Euler equation with periodic boundary
conditions and initial conditions F(p) cos p z + F(q) cos q z in the short-time
asymptotic regime. As has been shown numerically in W. Pauls et al., Physica D
219, 40-59 (2006), the type of the singularities depends on the angle between
the modes p and q. Here we show for the two particular cases of the angle going
to zero and to pi that the type of the singularities can be determined very
accurately, being characterised by the values 5/2 and 3 respectively. In these
two cases we are also able to determine the subdominant corrections.
Furthermore, we find that the geometry of the singularities in these two cases
is completely different, the singular manifold being located "over" different
points in the real domain.
