  We study surfaces in $\R^4$ whose tangent spaces have constant principal
angles with respect to a plane. Using a PDE we prove the existence of surfaces
with arbitrary constant principal angles. The existence of such surfaces turns
out to be equivalent to the existence of a special local symplectomorphism of
$\R^2$. We classify all surfaces with one principal angle equal to 0 and
observe that they can be constructed as the union of normal holonomy tubes. We
also classify the complete constant angles surfaces in $\R^4$ with respect to a
plane. They turn out to be extrinsic products. We characterize which surfaces
with constant principal angles are compositions in the sense of Dajczer-Do
Carmo. Finally, we classify surfaces with constant principal angles contained
in a sphere and those with parallel mean curvature vector field.
