Symmetrization of diagonalizable complex differential systems

Let L be a first order systemL(y,D)=ID0+∑j=1j=naj(y)Dj, where D0=∂/∂x0D0=∂/∂x0, Dj=∂/∂xjDj=∂/∂xj, y is a real vector parameter, I   is the idendity 3×33×3 matrix and aj(y)aj(y) is a 3×33×3 matrix-valued complex smooth function.Let L(y,ξ)L(y,ξ) be the symbol of L(y,D)L(y,D). We assume: ∀y, the real reduced dimension of L in y   is 5 and L(y,ξ)L(y,ξ) is symmetrizable: ∃T(y)∃T(y) such that: T−1(y)L(y,ξ)T(y)T−1(y)L(y,ξ)T(y) is hermitian ∀ξ  . We assume the nonexistence of some double characteristics depending on the reduced form of the system. Then: L(y,ξ)L(y,ξ) is smoothly symmetrizable ⟺∃T(y)∃T(y) smooth (same smoothness as the coefficients) such that: T−1(y)L(y,ξ)T(y)T−1(y)L(y,ξ)T(y) is hermitian ∀ξ.