**Construction 1** :
$A(-3,0,0), \quad B(3,0,0), \quad C(0,-3,\sqrt{7}), \quad D(0,3,\sqrt{7})$.
$|AB|=6, \; |AC|=5, \; |AD|=5, \; |BC|=5, \; |BD|=5, \; |CD|=6$.
When add point $E(0,0,0) \in AB$, then new tetrahedra $AECD$ and $BECD$ are integer too.
**Construction 2** :
$A\left(0,0,-\frac{577}{2}\right), \quad B\left(0,0,\frac{577}{2}\right), \quad C\left(\frac{\sqrt{1155}}{2}, 0, 0\right), \quad D\left(\frac{24\sqrt{579}}{\sqrt{1155}},\frac{3}{2\sqrt{1155}}, 0\right)$.
$|AB|=577, \; |AC|=289, \; |AD|=289, \; |BC|=289, \; |BD|=289, \; |CD|=24$.
When add points:
$E_{1,2}(0,0,\pm\frac{1}{2})$,
$E_{3,4}(0,0,\pm8\frac{1}{2})$,
$E_{5,6}(0,0,\pm15\frac{1}{2})$,
$E_{7,8}(0,0,\pm23\frac{1}{2})$,
$E_{9,10}(0,0,\pm39\frac{1}{2})$,
$E_{11,12}(0,0,\pm56\frac{1}{2})$,
$E_{13,14}(0,0,\pm95\frac{1}{2})$,
(which belong to $AB$),
then each tetrahedron $AE_jCD$, $BE_jCD$, where $j=1,2,...,14$, is integer tetrahedron too.