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.