Conserved quantities for polyhomogeneous spacetimes

The existence of conserved quantities with a structure similar to the Newman-Penrose quantities in a polyhomogeneous spacetime is considered. The most general form for the initial data formally consistent with the polyhomogeneous setting is found. The subsequent study is done for those polyhomogeneous spacetimes where the leading term of the shear sigma contains no logarithmic terms. It is found that for these spacetimes the original NP quantities cease to be constants, but it is still possible to construct a set of ten other quantities that are constant. From these quantities it is possible to obtain, as a particular case, the conserved quantity found by Chrusciel et al.