  The genus g of an F_{q^2}-maximal curve satisfies g=g_1:=q(q-1)/2 or g\le
g_2:= [(q-1)^2/4]. Previously, such curves with g=g_1 or g=g_2, q odd, have
been characterized up to isomorphism. Here it is shown that an F_{q^2}-maximal
curve with genus g_2, q even, is F_{q^2}-isomorphic to the nonsingular model of
the plane curve \sum_{i=1}^{t}y^{q/2^i}=x^{q+1}, q=2^t, provided that q/2 is a
Weierstrass non-gap at some point of the curve.
