We show how to build a {\em fully complete} model for the {\em maximal
theory} of the simply typed $\lambda$-calculus with $k$ ground
constants, $\lambda_k$. This is obtained by {\em linear realizability}
over an {\em affine combinatory algebra} of {\em partial involutions}
from natural numbers into natural numbers. For simplicitly, we give
the details of the construction of a fully complete model for
$\lambda_k$ extended with ground {\em permutations}. The fully
complete minimal model for $\lambda_k$ can be obtained by carrying out
the previous construction over a suitable subalgebra of partial
involutions.  The full completeness result is then put to use in order
to prove some simple results on the maximal theory.