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.  These metrics arose in   questions from  geometic mechanics.
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.