All materials are vitrified by fast quenching even monoatomic substances. Second melting temperatures accompanied by weak exothermic or endothermic heat are often observed at Tn+ after remelting them above the equilibrium thermodynamic melting transition at Tm. These temperatures Tn+ are due to the breaking of bonds (configurons) or antibonds depending on the thermal history which is explained by using a non-classical nucleation equation. Their multiple existence in monoatomic elements is now demonstrated by molecular dynamics simulations and still predicted. Proposed equations show that crystallization enthalpy is reduced at the temperature Tx due to new vitrification of non-crystallized parts and their melting at Tn+. These glassy parts being equal above Tx to singular values or to their sum are melted at various temperatures Tn+ and attains 100% in Cu46Zr46Al8 and 86.7% in bismuth. These first-order transitions at Tn+ are either reversible or irreversible depending on the formation of superatoms either solid or liquid.