  Let X be a smooth, complex Fano variety. For every prime divisor D in X, we
set c(D):=dim ker(r:H^2(X,R)->H^2(D,R)), where r is the natural restriction
map, and we define an invariant of X as c_X:=max{c(D)|D is a prime divisor in
X}. In a previous paper we showed that c_X<9, and that if c_X>2, then either X
is a product, or X has a flat fibration in Del Pezzo surfaces. In this paper we
study the case c_X=2. We show that up to a birational modification given by a
sequence of flips, X has a conic bundle structure, or an equidimensional
fibration in Del Pezzo surfaces. We also show a weaker property of X when
c_X=1.
