This talk is concerned with a family of Hardy-Sobolev doubly critical $p$-Laplace systems defined in $\mathbb{R}^N$, with $1<p<N$. In particular, we investigate qualitative properties of the solutions to the aforementioned system by employing the moving plane method. When $p \neq 2$, the application of this method becomes highly nontrivial due to the nonlinear nature of the $p$-Laplace operator. Moreover, adapting standard techniques encounters further obstructions caused by the Hardy potential. To overcome these difficulties, we employ a different test function method, which, in turn, introduces additional technical challenges. It is also worth emphasising that, due to the absence of the Kelvin transformation, further analytical tools must be developed and adopted.