We recast the Podleś spheres in the noncommutative physics context by showing that they can be regarded as slices along the time coordinate of the different regions of the quantum Minkowski space-time. The investigation of the transformations of the quantum sphere states under the left coaction of the ${\text{SO}}_q\left(3\right)$ group leads to a decomposition of the transformed Hilbert space states in terms of orthogonal subspaces exhibiting the periodicity of the quantum sphere states.

We solve the problem of description of nonsingular pairs of compatible flat metrics for the general $N$-component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. The integrating of these equations is based on reducing to a special nonlinear differential reduction of the Lamé equations and using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method).