  The interplay between quantum chaos and integrability has been extensively
studied in the past decades. We approach this topic from the point of view of
geometry encoded in the quantum geometric tensor, which describes the
complexity of adiabatic transformations. In particular, we consider two generic
models of spin chains that are parameterized by two independent couplings. In
one, the integrability breaking perturbation is global while, in the other,
integrability is broken only at the boundary. In both cases, the shortest paths
in the coupling space lead towards integrable regions and we argue that this
behavior is generic. These regions thus act as attractors of adiabatic flows
similar to river basins in nature. Physically, the directions towards
integrable regions are characterized by faster relaxation dynamics than those
parallel to integrability, and the anisotropy between them diverges in the
thermodynamic limit as the system approaches the integrable point. We also
provide evidence that the transition from integrable to chaotic behavior is
universal for both models, similar to continuous phase transitions, and that
the model with local integrability breaking quickly becomes chaotic but avoids
ergodicity.
