Problem setting: Let $x_i\in\mathbb{R}^d$ and $a_i\in [0,1]$, for all $i = 1,\dots, k$ ( with $k\geq d$), and define $$M(a) = \sum_{i = 1}^k a_i x_ix_i^T.$$

Question: Is there any closed-form solution (for $a$) to this set of equations?

$$\begin{cases}\text{trace}\left(M(a)^{-1}x_jx_j^T\right) = \text{trace}\left(M(a)^{-1}x_lx_l^T\right), \forall j \neq l,\\ \sum_{i =1}^k a_i= 1.\end{cases}$$

Related question: here.