  Previous numerical studies have revealed the existence of embedded solitons
(ESs) in a class of multi-wave systems with quadratic nonlinearity, families of
which seem to emerge from a critical point in the parameter space, where the
zero solution has a fourfold zero eigenvalue. In this paper, the existence of
such solutions is studied in a three-wave model. An appropriate rescaling casts
the system in a normal form, which is universal for models supporting ESs
through quadratic nonlinearities. The normal-form system contains a single
irreducible parameter $\epsilon $, and is tantamount to the basic model of
type-I second-harmonic generation. An analytical approximation of WKB type
yields an asymptotic formula for the distribution of discrete values of
$\epsilon $ at which the ESs exist. Comparison with numerical results shows
that the asymptotic formula yields an exact value of the scaling index, -6/5,
and a fairly good approximation for the numerical factor in front of the
scaling term.
