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Using the intertwining matrix of the IRF-Vertex correspondence we propose adeterminant representation for the generating function of the commutingHamiltonians of the double elliptic integrable system. More precisely, it is aratio of the normally ordered determinants, which turns into a singledeterminant in the classical case. With its help we reproduce the recentlysuggested expression for the eigenvalues of the Hamiltonians for the dual toelliptic Ruijsenaars model. Next, we study the classical counterpart of ourconstruction, which gives expression for the spectral curve and thecorresponding $L$-matrix. This matrix is obtained explicitly as a weightedaverage of the Ruijsenaars and/or Sklyanin type Lax matrices with the weightsas in the theta function series definition. By construction the $L$-matrixsatisfies the Manakov triple representation instead of the Lax equation.Finally, we discuss the factorized structure of the $L$-matrix.

Characteristic determinant and Manakov triple for the double elliptic integrable system