Representations of algebra of harmonic eiconals

We describe the spectrum of the sub-algebra $\mathscr{E}$ of bounded operators on the space $H$ of potential harmonic vector fields on the disk $\mathbb{D}$ generated by the operator integrals (eiconals) of the form $\int t dP_{\Gamma_t}$, where $t\mapsto\Gamma_t$ is an expanding family of arcs in $\mathbb{T}:=\partial\mathbb{D}$ and $P_{\Gamma_t}$ is a projection on the subspace of $H$ spanned by vector fields normal to $\mathbb{T}\setminus\Gamma_t$.