  We develop the technique of the variational approximation for solitons in two
directions. First, one may have a physical model which does not admit the usual
Lagrangian representation, as some terms can be discarded for various reasons.
For instance, the second-harmonic-generation (SHG) model considered here, which
includes the Kerr nonlinearity, lacks the usual Lagrangian representation if
one ignores the Kerr nonlinearity of the second harmonic, as compared to that
of the fundamental. However, we show that, with a natural modification, one may
still apply the variational approximation (VA) to those seemingly flawed
systems as efficiently as it applies to their fully Lagrangian counterparts. We
call such models, that do not admit the usual Lagrangian representation,
\textit{semi-Lagrangian} systems. Second, we show that, upon adding an
infinitesimal tail that does not vanish at infinity, to a usual soliton ansatz,
one can obtain an analytical criterion which (within the framework of VA) gives
a condition for finding \textit{embedded solitons}, i.e., isolated truly
localized solutions existing inside the continuous spectrum of the radiation
modes. The criterion takes a form of orthogonality of the radiation mode in the
infinite tail to the soliton core. To test the criterion, we have applied it to
both the semi-Lagrangian truncated version of the SHG model and to the same
model in its full form. In the former case, the criterion (combined with VA for
the soliton proper) yields an \emph{exact} solution for the embedded soliton.
In the latter case, the criterion selects the embedded soliton with a relative
error $\approx 1%$.
