A generalized quadrangle $\cal S$ is laxly embedded
in a (finite) projective space {\bf PG}$(d,q)$ if
$\cal S$ is a subgeometry of the geometry of points
and lines of {\bf PG}$(d,q)$, with the only condition
that the points of $\cal S$ generate the whole space
{\bf PG}$(d,q)$ (which one can always assume without
loss of generality). In this paper, we classify thick
laxly embedded quadrangles satisfying some additional
hypotheses. The hypotheses are (a combination of) a
restriction on the dimension $d$, a restriction on the
parameters of $\cal S$, and an assumption on the
isomorphism class of $\cal S$. In particular, the
classification is complete in the following cases:
\begin{enumerate}
\item[(1)] for $d\geq 5$;
\item[(2)] for $d=4$ and $\cal S$ having `known' order
$(s,t)$ with $t\not= s^2$;
\item[(3)] for $d\geq 3$ and $\cal S$ isomorphic to a finite
Moufang quadrangle distinct from $W(s)$ with $s$ odd.
\end{enumerate}
As a by-product, we obtain a new characterization
theorem of the classical quadrangle $H(4,s^2)$, and we
also show that every generalized quadrangle of order
$(s,s+2)$, with $s>2$, has at least one non-regular line.
2000 Mathematics Subject Classification: 51E12.
