By writing the delta function as the color-singlet state of the canonical ensemble for the unimodular-like U ( N$_{c}$ ) (i.e. SU ( N$_{c}$ )) reads B. Orthogonal symmetry group O$_{(}$$_{S}$$_{)}$ ( N ) The unitary Hermitian system may be decomposed into symmetric Hermitian and anti- symmetric Hermitian real orthogonal components. The orthogonal symmetry group is sym- metric one and is invariant under the time-reversal transformation. The unitary Hermitian system can be fully or partially projected into (symmetric) orthogonal Hermitian system. The projection of the unitary Hermitian system into the real symmetric orthogonal system is very relevant to the quark-gluon plasma under extreme conditions such as the hot nuclear matter. The transformation matrix, namely, U of the orthogonal group is a symmetric one. It has N 2 elements and these elements are restricted to N + N ( N − 1) / 2 conditions. These constraints reduce the number of the adjoint variables to N$_{adj}$ = N ( N − 1) / 2 while the num- ber of the fundamental variables remain the same N$_{fun}$ = N in the orthogonal symmetry group. The invariance Haar measure for the orthogonal symmetry group reads The normalisation constant N$_{U}$ is determined by When the saddle points accumulate around the origin, the invariance Haar measure becomes 23 ∫ dµ$_{O}$$_{(}$$_{S}$$_{)}$ ( θ$_{1}$ · · · θ$_{N}$ ) e − 1 $_{2}$a ∑ N i $_{=1}$θ 2 i = N$_{O}$ ∫ ∞ −∞ N ∏ i =1 dθ$_{i}$ 2 π N ∏ i<j | θ$_{i}$ − θ$_{j}$ | e − 1 $_{2}$a ∑ N i $_{=1}$θ 2 i . (75) N$_{O}$ ∫ π − π N ∏ i =1 dθ$_{i}$ 2 π N ∏ i<j ∣ 2 sin ( θ$_{i}$ − θ$_{j}$ 2 ) ∣ ∣ = 1 . (74) ∫ dµ$_{O}$$_{(}$$_{S}$$_{)}$ ( θ$_{1}$ · · · θ$_{N}$ ) = N$_{O}$ ∫ π − π N ∏ i =1 dθ$_{i}$ 2 π N ∏ i<j ∣ sin ( θ$_{i}$ − θ$_{j}$ 2 ) ∣ ∣ . (73) ∫ dµ$_{SU}$ ( θ$_{1}$ · · · θ$_{N}$ ) e − 1 $_{2}$a ∑ N i $_{=1}$θ 2 i = √ 2 πa N ∫ dµ$_{U}$ ( θ$_{1}$ · · · θ$_{N}$ ) e − 1 $_{2}$a ∑ N i $_{=1}$θ 2 i . (72) δ ( N ∑ i θ$_{i}$ ) = ∫ ∞ −∞ dx exp ( i x N ∑ i θ$_{i}$ ) , (71)