This equipment comprises an iliac rod (2), a connection assembly (3, 4) for connecting that iliac rod to a vertebral bar, an iliac anchoring member and an iliac part connecting the iliac rod to that iliac anchoring member; said connecting assembly (3, 4) is independent of the iliac rod (2) and comprises two portions, namely: —a first portion (3), for connecting to said vertebral bar (50), that forms a closed conduit (10) for engagement of that portion (3) on that bar (50); and —a second portion (4), for connecting to said iliac rod (2), that forms a conduit (12) for the engagement of that second portion (4) on the iliac rod (2). According to the invention, —said first portion is formed by a first connecting part (3) and said second portion is formed by a second connecting part (4), said first and second connecting parts (3, 4) being independent of one another; said first connecting part (3) comprises a portion (3a) that forms said closed conduit (10) and a threaded rod (3b) protruding from said portion (3a); said second connecting part (4) forms a passage (11) through it for the engagement of said threaded rod (3b) of the first connecting part (3), with a pivoting capacity of said second connecting part (4) relative to said first connecting part (3).