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Using the transfer matrix method for Axial Next-Nearest Neighbor Ising modelwithout an external field on a closed chain of spins of width 2 in thedirection of interaction of only nearest neighbors and length L in thedirection of interaction of nearest neighbors and next nearest neighbors, wefound exact values of the partition function in a finite strip of length L ,free energy, internal energy per node, heat capacity in a finite strip oflength L and in the thermodynamic limit. The entire spectrum of the 16x16transfer matrix and the structure of all transfer matrix eigenvectors arefound. The problem of finding the spectrum of the transfer matrix is reduced tofinding the spectrum of two matrixes of size 8x8. The characteristic polynomialof the first matrix is decomposed into two linear polynomials of multiplicitytwo and two square trinomials. The characteristic polynomial of the secondmatrix is decomposed into two polynomials of the fourth degree, the roots ofwhich are found by the Ferrari method. The free energy in the thermodynamiclimit has a very simple form and is expressed in terms of the logarithm of theroot of the quadratic equation. Two theorems on the physical characteristics ofthe ANNNI model in a finite closed strip of length L and in the thermodynamiclimit are formulated and proved. An example of calculation is given for somevalues of the model parameters in the thermodynamic limit, heat capacity graphsare plotted. A comparative analysis of the heat capacity for thedouble-stranded ANNNI model and the ANNNI model on the entire flat lattice isgiven. The closest connection and similarity of the physical characteristics ofthe two-chain model and the model on the entire flat lattice is shown.

The exact solution of the Axial Next-Nearest-Neighbor Ising model on width 2 spin chains