  The purpose of this paper is to study basic property of the operator
  $$\mathcal{L}_{\mu_1,\mu_2} u=-\Delta +\frac{\mu_1 }{|x|^2}x\cdot\nabla
+\frac{\mu_2 }{|x|^2},$$ which generates at the origin due to the critical
gradient and the Hardy term, where $\mu_1,\mu_2$ are free parameters. This
operator arises from the critical Caffarelli-Kohn-Nirenberg inequality. We
analyze the fundamental solutions in a weighted distributional identity and
obtain the Liouville theorem for the Lane-Emden equation with that operator, by
using the classification of isolated singular solutions of the related Poisson
problem in a bounded domain $\Omega \subset \mathbb{R}^N$ ($N \geq 2$)
containing the origin.
