A vector model describing oil-water-amphiphile mixtures, originally proposed by Ciach, Hoye and Stell, is investigated on the simple cubic lattice. Our previous approach, involving an exact integration over the amphiphile orientational degrees of freedom, is reformulated to consider the more general case of asymmetric amphiphiles. The resulting spin-1 effective Hamiltonian, containing many-site temperature-dependent interactions, is then treated in a local mean field approximation and the phase behavior at finite temperature is extensively analyzed. In order to stress asymmetry effects, a detailed comparison with the symmetric case is performed.