Background
Define essential space as $$\varepsilon=\{E \in \mathbb R^{3\times3}|E=\hat{T}R\}$$ $$\hat{T}\in\{S\in \mathbb R^{3\times3}|S^T=-S\}$$ $$R\in\{A\in\mathbb R^{3\times3}|A^TA=I,\det(A)=1\}$$ that is any matrix $E \in \varepsilon$ is skew-symmetric matrix $\hat T$ post-multiplied by a rotation matrix R.
According to this paper, $E$ resides in $\varepsilon$ if and only if $$E=U\Sigma V^T,\Sigma = diag(\sigma,\sigma,0).$$
Optimization Problem
For a matrix $F \in \mathbb R^{3\times 3}$, searching it's projection on the essential space can be considered as solving following optimization problem: $$\arg\min_{E} \|F-E\|_F^2 $$ $$s.t. \qquad E \in \varepsilon.$$ Suppose $F=Udiag(\lambda_1,\lambda_2,\lambda_3) V^T$, then solution to above problem is $$E = Udiag(\sigma,\sigma,0) V^T, \sigma = \frac{\lambda_1+\lambda_2}{2}$$
Question
Let $E=U_1diag(\sigma,\sigma,0)V_1^T$ be any matrix in $\varepsilon$. I have derived above solution if $U_1=U,V_1=V$, but stuck on cases in which $U_1\ne U$ or $V_1 \ne V$.