On the Dirichlet problem for the harmonic vector fields equation

We show that any harmonic (with respect to the Bergman metric) vector field tangent to the Levi distribution of the foliation by level sets of the defining function φ(z)=−K(z,z)−1/(n+1) of a strictly pseudoconvex bounded domain Ω⊂Cn which is smooth up to the boundary must vanish on ∂Ω. If n≠5 and uT is a harmonic vector field with u∈C2(Ω¯) then u|∂Ω=0.