I am rusty on differential equations. My application requires that I characterize the function $S:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $$ \sum_{j=1}^{n}x_{j}\frac{\partial}{\partial{x_{i}}}S_{j}(x)=0\mbox{ for all }i=1,\ldots,n $$ where $S_{j}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ assigns the $j$-th component of $S(x)$ to $x$. I know one function which fulfills this constraint: $$S_{j}(x)=\left(\ln\sum{}x_{k}\right)-\ln{}x_{j}$$
Are there others?