Lagrangian with Infinite solutions?

Question

I am trying to minimize the quantity $$\frac{1}{2}\left(\sum_{i=1}^k x_i^2\right) + x_Ax_B$$ where $A$ and $B$ are fixed such that $1\leq A<B\leq k$. We have the constraint $$\sum_{i=1}^k x_i=1$$ and non-negativity constraints on each $x_i$. I know that if I use Lagrange multipliers (we have continuous partial derivatives), I will need to check the boundary / endpoints, but by the nature of the problem I am working on, that will not be too hard. Anyways, I have the first order conditions: \begin{align*} x_A+x_B+\lambda &= 0\\ x_i+\lambda &= 0\quad\text{ for $i\neq A,B$} \end{align*} Plugging into the constraint using $x_A+x_B=-\lambda=x_i$ and simplifying gives me $$(k-1)(x_A+x_B)=1\implies \lambda=-\frac{1}{k-1}$$ Here is where I am confused. It seems to me that $x_A$ and $x_B$ can take on any values; it only matters that they sum up to $-\frac{1}{k-1}$. I've never really seen a Lagrangian give a solution with a free variable, so I am wondering if it is valid.