The (orbifold) space of symmetric complex matrix 3 - simplified

Question

Let $K$ be a rank-2 symmetric complex matrix, such that the transpose $K^T=K$ is itself.

Modifying the previous attempt, let $V$ be a rank-2 matrix in the special unitary matrix, $$V \in SU(2) .$$ (There is a Autonne-Takagi theorem helping for the case however only for $V \in U(2)$, see the linked answer.)

Consider the identification between any K and K' of any rank-2 symmetric complex matrix, $$ K\sim K', $$ if it satisfies $$ V^T K V =K', $$ for any $V \in SU(2) .$

question:

1. What is the real dimension of the new space of $K$ (under the $K\sim K'$ and $V^T K V =K'$, for any $V \in SU(2) $ condition)?

2. How do we parametrize this new space of $K$ in terms of a rank-2 matrix (mod out the redundancy under the $K\sim K'$ and $V^T K V =K'$, for any $V \in SU(2) $ condition)?

(p.s. This space may be a called an orbifold space(?). i.e. The (orbifold) space of symmetric complex matrix after mod out a relation identifying a unitary matrix.)