If $n\geq 2$, generically, you are unlikely to encounter such an equation with only one solution.

Let $d$ be the dimension of our symmetric matrices.

$\bullet$ Consider the case $n=2$. Then the algebraic resolution of the system leads to obtaining the roots of a real polynomial $P$ of degree $2^d$ (generically). Our system admits a unique solution iff $P$ admits only one real root. Since $degree(P)$ is even, this is not possible.

Then we must assume that $n\geq 3$.

$\bullet$ Note that a random polynomial of degree $p$ has

$O(\log(p))$ real zeros on average -when the coefficients are independent standard normals-

or $O(\sqrt{p})$ -when the variance varies with the index of the coefficient-.

If $A_1,A_2,K$ are randomly chosen, then $P$ can be roughly considered as random and the probability that $P$ admits a single real root is very small when $p$ is large.

$\bullet$ For example, if $d=2,n=3$, we obtain -generically- a polynomial of degree $11$. I did some tests and got $5,7,9$ or $11$ real roots.

Finally, the existence of a unique solution is a very special case.