The paper is devoted to solving the problem of a cross-seam bellows made of a shape memory alloy (SMA) under the an axial load during a direct thermoelastic martensitic phase transformation. As special cases, two different approaches to accounting for the operator associated with the Poisson’s coefficient during the Laplace transform are considered. As a result, a hypothesis is put forward that it is possible not to take into account the operator associated with the Poisson’s coefficient during the Laplace transformation in the case of considering incompressible materials, but to consider this coefficient as a material parameter. which was verified numerically in the framework of this work. The bellows behavior was described in the framework of the model of linear deformation of the SMA during phase transformations and was modeled as the behavior of a system of ring plates. The problem was solved within the framework of an unrelated formulation, the distribution of the phase composition and temperature parameter over the bellows material at each moment of time was assumed to be uniform. Similarly, the possibility of structural transformation in the bellows material, the variability of elastic modules during the phase transition, and the property of the SMA’s resistance to diversity were neglected. To obtain an analytical solution of all the equations of the boundary value problem, the Laplace transform method was used in terms of the volume fraction of the martensitic phase. After the transformation in the image space, an equivalent elastic problem is obtained. When solving this problem, the Laplace images of the desired quantities are obtained in the form of analytical expressions that include operators that are Laplace images of elastic constants. These expressions are fractional-rational functions of the Laplace image of the phase composition parameter. To return to the original space, the expressions for the desired values in the image space are decomposed into simple fractions. As a result of the inversion of these fractions, the desired analytical solutions are obtained.