[en] The equation of the Bohr–Mottelson Hamiltonian with a sextic oscillator potential is solved for $\mathrm{\gamma <!--\; ??\; -->}$-rigid prolate nuclei. The associated shape phase space is reduced to three variables which are exactly separated. The angular equation has the spherical harmonic functions as solutions, while the $\mathrm{\beta <!--\; ??\; -->}$ equation is converted to the quasi-exactly solvable case of the sextic oscillator potential with a centrifugal barrier. The energies and the corresponding wave functions are given in closed form and depend, up to a scaling factor, on a single parameter. The $0^{+}$ and $2^{+}$ states are exactly determined, having an important role in the assignment of some ambiguous states for the experimental $\mathrm{\beta <!--\; ??\; -->}$ bands. Due to the special properties of the sextic potential, the model can simulate, by varying the free parameter, a shape phase transition from a harmonic to an anharmonic prolate $\mathrm{\beta <!--\; ??\; -->}$-soft rotor crossing through a critical point. Numerical applications are performed for 39 nuclei: ${}^{98-<!--\; ???\; -->108}$Ru, ${}^{100,102}$Mo, ${}^{116-<!--\; ???\; -->130}$Xe, ${}^{132,134}$Ce, ${}^{146-<!--\; ???\; -->150}$Nd, ${}^{150,152}$Sm, ${}^{152,154}$Gd, ${}^{154,156}$Dy, ¹⁷²Os, ${}^{180-<!--\; ???\; -->196}$Pt, ¹⁹⁰Hg and ²²²Ra. The best candidates for the critical point are found to be ¹⁰⁴Ru and ${}^{120,126}$Xe, followed closely by ¹²⁸Xe, ¹⁷²Os, ¹⁹⁶Pt and ¹⁴⁸Nd. (paper)