  Let $K$ be a global field of characteristic different from 2 and $u(x)\in
K[x]$ be an irreducible polynomial of even degree $2g\ge 6$, whose Galois group
over $K$ is either the full symmetric group $S_{2g}$ or the alternating group
$A_{2g}$. We describe explicitly how to choose (infinitely many) pairs of
distinct elements $t_1, t_2$ of $K$ such that the $g$-dimensional jacobian of a
hyperelliptic curve $y^2=(x-t_1)(x-t_2))u(x)$ has no nontrivial endomorphisms
over an algebraic closure of $K$ and has big $\ell$-adic monodromy.
