For MBHE-HL, Bertoli et al. performed a character analysis for the roots 𝜆 𝑗 (𝑗 = 0,1,2 positive real root and the other two, both negative real ones, with the (unconfirmed) possibility of a double real root or complex conjugate, while for the (practical) region 10 -10 < 𝜉 𝑞 ≤ 100 (𝑞 = 1 -4), it was found that the roots are simple real ones. In the next section, these results will be used to establish -by analogy -the character of the roots for the solution of the TTHE-HL model.