Directed harmonic currents near non-hyperbolic linearizable singularities

Let $(\mathbb{D}^2,\mathcal{F},\{0\})$ be a singular holomorphic foliation on the unit bidisc $\mathbb{D}^2$ defined by linear vector field \[ z \,\frac{\partial}{\partial z}+ \lambda \,w w}, \] where $\lambda\in\mathbb{C}^*$. Such has non-degenerate linearized singularity at $0$. $T$ harmonic current directed $\mathcal{F}$ which does not give mass to any of two separatrices $(z=0)$ and $(w=0)$ whose trivial extension $\tilde{T}$ across $0$ is $dd^c$-closed. The Lelong number describes distribution foliated space. In 2014 Nguyen proved that when $\lambda\notin\mathbb{R}$, i.e. hyperbolic singularity, vanishes. For non-hyperbolic case $\lambda\in\mathbb{R}^*$ article proves following results. $0$: 1) strictly positive if $\lambda>0$; 2) vanishes $\lambda\in\mathbb{Q}_{<0}$; 3) $\lambda<0$ invariant under action some cofinite subgroup monodromy group.