Are you sure that your model setup has a solution? The unit root seems like the type of problem that occurs in open economy setup models. Essentially, after a shock that affects debt, the permanent income hypothesis kicks in and the household just consumes the annuity out of its assets forever, never decumulating debt. It could now be that if you provide access to the bond market for the second household, it is set on a trajectory of permanently accumulating/decumulating assets, because the interest rate does not react. In open economy models there is usually a debt-elastic interest rate premium or something similar (or alternatively a borrowing constraint that kicks in via a Lagrange multiplier in the Euler equation)