Exact solutions of a variant Boussinesq system

We apply the symmetry method based on the Fréchet derivative of the differential operators to deduce the Lie symmetries of the following variant of the Boussinesq equationsut+α1(t)vx+β1(t)uux+γ1(t)uxx=0vt+α2(t)uvx+β2(t)vux+γ2(t)vxx+p(t)uxxx=0where αi(t), βi(t), γi(t), i = 1, 2 and p(t) are arbitrary functions of t. For each infinitesimal generator in the optimal system of subalgebras we study the reduced ODE and, among other solutions, furnish some nontrivial exact solutions in terms of hyperbolic functions.