On Martin Boundaries for Invariant Markov Processes on a Solvable Group

We consider left-invatiant diffusion processes on the group $G=\Bigl\{\begin{Vmatrix}x&y\\0&1\end{Vmatrix},\ x>0\Bigr\}$ and find all minimal positive solutions of the equation $\widehat L_a=0$ where $L_a$ is some leftinvariant differential operator of the second order on $G$. These results are different from those in the cases of abelian and nilpotent groups where all minimal harmonic functions are non-negative characters.