An alternative to the Simon tensor

The Simon tensor gives rise to a local characterization of the Kerr-NUT family in stationary class vacuum spacetimes. We find that symmetric and traceless quotient space Killing trajectory offers useful alternative tensor. Our is distinct from spatial dual illustrates geometric property three dimensional more manifest. reconstruction procedure metric for which generalized vanishes spelled out detail. give four description this terms Coulomb part imaginary selfdual Weyl tensor, corresponds generalization three-index defined by Mars. This allows us establish new simple criterion family: gradient Ernst potential becomes non-null eigenvector also discuss ${\rm SU}(1,2)$ covariant extension obstruction into Einstein-Maxwell system as an intrinsic Kerr-Newman-NUT family.