**Abstract** : Extensive numerical solutions of the hypernetted-chain (HNC) and Rogers-Young (RY) integral equations are presented for the pair structure of a system of two coupled replicae (1 and 2) of a ``soft-sphere'' fluid of atoms interacting via an inverse-12 pair potential. In the limit of vanishing inter-replica coupling epsilon(12), both integral equations predict the existence of three branches of solutions: (1) A high temperature liquid branch (L), which extends to a supercooled regime upon cooling when the two replicae are kept at epsilon(12) = 0 throughout; upon separating the configurational and vibrational contributions to the free energy and entropy of the L branch, the Kauzmann temperature is located where the configurational entropy vanishes. (2) Starting with an initial finite coupling epsilon(12), two ``glass'' branches G(1) and G(2) are found below some critical temperature, which are characterized by a strong remnant spatial inter-replica correlation upon taking the limit epsilon(12) -> 0. Branch G(2) is characterized by an increasing overlap order parameter upon cooling, and may hence be identified with the hypothetical ``ideal glass'' phase. Branch G(1) exhibits the opposite trend of increasing order parameter upon heating; its free energy lies consistently below that of the L branch and above that of the G(2) branch. The free energies of the L and G(2) branches are found to intersect at an alleged ``random first-order transition'' (RFOT) characterized by weak discontinuities of the volume and entropy. The Kauzmann and RFOT temperatures predicted by RY differ significantly from their HNC counterparts.