Non-vacuum metrics for the Newman-Unti-Tamburino background: A coordinate-free approach to diverging and twisting solutions

The geometry of the Newman-Unti-Tamburino (NUT) vacuum solution is characterized as the unique Petrov Type D vacuum metric such that the two double principal null directions form an integrable distribution. We study expanding and twisting non-vacuum Type D metrics in this geometry, with the additional assumption ${\mathrm{\Phi }}_{01}={\mathrm{\Phi }}_{12}=0$. We prove that these conditions determine the solutions up to a freedom in ${\mathrm{\Phi }}_{11}\pm 3\mathrm{\Lambda }$.