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

I am trying to solve the following set of equations (for $\alpha$)

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

Is there any closed-form solution to this set of equations?