  We study the modified $J$-flow introduced in [15], particularly the
singularities of the flow using the Calabi symmetry. In [20], on toric
manifolds the convergence of modified $J$-flow to the smooth solution was
proven under the assumption of positivity of certain intersection numbers. In
the case of the Calabi ansatz, we show that if some of those intersection
numbers are not positive, then the modified $J$-flow blows up along some
variety, and away from the variety we prove the convergence to the solution.
  As in [10], we also prove that the convergence behavior of the modified
$J$-flow with Calabi symmetry depends on the topological constants $c$ and the
minimum of the Hamiltonian function.
