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The puzzle of finding consistent nuclear configurations for explaining boththe decay probabilities and moments of the ${9/2}^-$, ${8}^+$ and ${21/2}^-$isomers in and around $N=126$ closed shell has been approached in thegeneralized seniority scheme. Though $h_{9/2}$ is the dominant orbital nearFermi energy, the role of configuration mixing from the surrounding $f_{7/2}$and $i_{13/2}$ orbitals is found to be very important for the consistentexplanation of all the isomeric properties such as the $B(E2)$ rates,$Q-$moments and $g-$factors. The structural behavior of the closed shell$N=126$ isotonic isomers turns out to be very similar to the $N=124$ and$N=128$ isotonic isomers which have two neutron-holes and twoneutron-particles, respectively. This is due to the pairing symmetries ofnuclear many-body Hamiltonian. As a confirmation, the microscopic shell modeloccupancies are also calculated for these isomers in $N=126$ chain whichsupport the generalized seniority results. Additional arguments using thesystematics of odd-proton ${9/2}^-$ states in Tl ($Z=81$), Bi ($Z=83$), At($Z=85$) and Fr ($Z=87$) isotopes are also presented.

A puzzle on isomeric configurations in and around $N=126$ closed shell