Let (M, T{sub 1,0} (M)) be a CR manifold (of hypersurface type) where (1) T{sub 1,0}(M) denotes its CR structure (of CR dimension N). Let {mu} be a pointwise C-anti-linear endomorphism of T{sub 1,0}(M). Let (T{sub 1}, ..., T{sub N}) be a (local) frame of T{sub 1,0}(M) and consider the first order PDE (with variable coefficients): T{sub J}{line_integral} = {mu}{sub j}{sup l}T{sub l}{line_integral} where {mu}T{sub j} = {mu}{sub j}{sup l}T{sub l}. This is the Beltrami equation, cf. the terminology of A. Koranyi & H.M. Reimann. Next, assume that P {improper_subset} K er {mu}, so that {mu} descends to an endomorphism of H. Finally, if we restrict ourselves to basic unknown functions {line_integral} {element_of} {Omega}{sub B}{sup 0} (F) i.e. {line_integral} is constant along each leaf of (F) then (1) may be written as (2) {zeta}{sub {alpha}}({line_integral}) = {mu}{sub {alpha}}{sup {beta}} {zeta}{sub {beta}} ({line_integral}). This makes sense for an arbitrary CR foliation (F), of a C{sup {infinity}} manifold M, endowed with a C-anti-linear endomorphism {mu} of its transverse CR structure, and is invariant under a change of admissible frame. We refer to (2) as the (transverse) Beltrami equation of (M, F). We use the theory of CR foliations to show that themore » components of an automorphism {phi} preserving the transverse contact structure of a given embedded strictly pseudoconvex CR foliation satisfy (2) where {mu} is the complex dilatation of {phi} and conversely. For transversally Heisenberg CR foliations we use the results in sections 2 and 3 to characterize K-quasiconformality of a foliation automorphism. 10 refs.« less