The boundary integral equation with convection is derived for the symmetric Langer and Turski phase transformation model (Langer in Acta Metall 25:1113–1119, 1977). A linear morphological stability of the planar interface in the moving melt is studied. The stationary solution, dispersion relation and neutral stability surface are obtained using the perturbation theory. The present theory extends the theory by Langer and Turski with allowance for the plane-parallel flow of one liquid relative to another one. © 2023, The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature.