The self-dual Yang-Mills theory is investigated with the help of a new conformally covariant linear system, where the spectral parameter is a projective twistor [Formula: see text]. We derive from this linear system conformally covariant families of β-planes, on which the potential Aµ(x) is a pure gauge. They are parametrized by a dual projective twistor [Formula: see text], orthogonal to the spectral twistor Λ. Conformally covariant infinitesimal Bäcklund transformations (B.T.) are constructed for the gauge group [Formula: see text] or [Formula: see text], and for SU (N). They are characterized by (1) a Lie-algebra index 1≤a≤ dim g; (2) the spectral twistor Λ; (3) a second twistor index 1≤α≤4, (independent of Λ); (4) an arbitrary (analytic) function of the two independent solutions of the free linear system (Aµ=0). The algebra of these infinitesimal B.T. is computed. It turns to close up to a field-dependent gauge transformation, which vanishes for equal twistor indices. The reduction of the number of components of Λ to a single projective parameter [Formula: see text] leads to a loop algebra. In general it yields an infinite-dimensional algebra with five indices.