Abstract: NCCC is by no means a popular tool in physics, and certainly the possibilities of its application have not been fully exhausted. The spinor equivalent of NCCC appears in the literature as a local twistor connection. Thus, using NCCC, spacetimes admitting covariantly constant twistor fields were once found and identified as conformal metrics constructed earlier by Fefferman and used in the geometric approach to the Cauchy-Riemann structures. A new application of NCCC is the construction of conformally invariant symplectic potential on the space of Bach flat geometries. These include all Einstein spacetimes. NCCC provides new charges and currents well-defined both at points in spacetime and at its conformal boundary. Relevant examples will be demonstrated in the Graham-Fefferman coordinates.