  We obtain an asymptotic formula for the number of
$\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of irreducible binary
quartic forms with integer coefficients with vanishing $J$-invariant and whose
Hessians are proportional to the squares of reducible or positive definite
binary quadratic form. These results give a case where one is able to count
integral orbits inside a relatively open real orbit of a variety closed under a
group action of degree at least three.
