<p>A theory of the normal and superconducting states of piezomagnetic metals, which are altermagnetic materials, has been developed. This has been done in comparison with the corresponding theoretical description for metals that do not have spatial inversion symmetry. Particular attention has been paid to the problem of the anomalous Hall effect, and its absence has been demonstrated in both altermagnetic and non-centrosymmetric metals. It has been shown that the superconducting state in altermagnets always arises by pairing electrons from zones split by spin–orbit interaction, which makes the existence of such superconductors unlikely.</p>