Take the product $S^2 \times S^2$ of two two-spheres,
but perturb the product metric as follows.
Think of each $S^2$  as the unit two-sphere in   Euclidean 3-space
in the standard way 
so that for   $p \in S^2$ the tangent space
 $T_p S^2 = p^{\perp}$ is the two-plane in
3-space perpindicular to the unit vector p.
Then the standard round metric on  the unit $S^2$  is given by
$<v, v>_p = v \cdot v$ for $v \in T_p S^2$ where the dot product is
the standard dot product of 3-space. 
Now consider $P = (p_1, p_2) \in S^2 \times S^2$
and corresponding  $(v_1, v_2) \in T_{p_1} S^2 \times T_{p_2} S^2$
Declare 
$$|(v_1, v_2) |_p ^2 = A v_1 \cdot v_1   +  B v_1 \cdot v_2  + Cv_2 \cdot v_2 $$
for $A, B, C$ constants.   This quadratic form is   positive definite 
and so  defines a Riemannian metric on the product of the spheres provided (I guess) that  
$B^2 < 4AC$ and $A, C > 0$, and in any case,  certainly whenever  $B$ 
is small enough 
relative to $A, C > 0$.  

Question 1.  Does anyone have  a name for this family of
metrics on $S^2 \times S^2$? A reference for that name? 

When the cross term $B = 0$
this metric is the product metric of two `round' two-spheres whose radii squared are
$A$ and $C$ so the isometry group of the metric is the product $O(3) \times O(3)$ and  its geodesic flow is integrable.  Taking $B \ne 0$ breaks the symmetry  to  the diagonal $O(3)$ sitting inside the product of $O(3)$'s and it appears (from numerical experiments, expectations,  and, perhaps, lack of imagination) that
the geodesic flow of the metric is NOT integrable.  
 

Question 2.    When $B \ne 0$ can you prove that
the   geodesic flow is  not integrable?

Motivations.  Holger Dullin came up with this family of metrics while we were working on some   questions arising in   geometic mechanics and symplectic reduction.
The dynamics of a free spherical pendulum for example  -- two rods joined by a 
``spherical  joint'' and  thrown  into space with no gravity -- can be
put into the form of the geodesic equations for such a metric.