This question is closely related to the following MO question http://mathoverflow.net/questions/185126/characterizing-the-real-analytic-eisenstein-series


Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed with its Poincare metric. Let $w\in\mathbf{Z}$ be a weight and define the "weight $w$ Laplacian" on $\mathfrak{h}$ by 
$$
\Delta[w]=-y^2(\partial_x^2+\partial_y^2)+i\cdot w\cdot y\cdot\partial_x.
$$
Here $i=\sqrt{-1}$. In general, $\Delta[w]$ will commute with the $|_{w}$ right action of $GL_2^+(\mathbf{R})$ on Maass forms as is explained on page 130 of Bump's book on automorphic forms and representations. Consider the differential equation
$$
(1)\;\;\;\;\;\;\;\; \Delta[w] f(y)=(s-w/2)(1-s+w/2)f(y),
$$
where $f(y)$ is a function depending only on $y=Im(z)$. Solving the corresponding linear ODE of order $2$ in $y$ we see that the solution space of (1) (for $s\neq \frac{1}{2}$) has the form
$$
A(s)y^{s-w/2}+B(s)y^{1-s+w/2}.
$$


Let $g\in\mathbf{Z}_{\geq 1}$ be a fixed integer and consider the symmetric space $\mathfrak{h}^g$. Let $\underline{z}=(z_j)_{j=1}^g\in\mathfrak{h}^g$ where $z_j=x_j+iy_j$. Let $\underline{w}=(w_1,w_2,\ldots,w_g)\in\mathbf{Z}^g$ be an integral weight vector. We let $D_j[w_j]$ be the weight $w_j$ Laplacian with respect to the variable $z_j$.

Consider now the following differential system:
$$
(2)\;\;\;\;\;\;\;\;\; D_j[w_j]f(\underline{y})=(s-w_j/2)(1-s+w_j/2)f(\underline{y})\;\;\;\;\;\;\;\;\; (1\leq j\leq g)
$$
where $f(\underline{y})$ is a function depending only on $Im(\underline{z})=\underline{y}=(y_1,\ldots,y_g)$. A natural set of solutions for $(2)$ is given by 
$$
(3)\;\;\;\;\;\;\;\;\prod_{j=1}^g (A_j(s)y_j^{s-w_j/2}+B_j(s)y_j^{1-s+w_j/2})
$$


**Question 1:** Are all the solutions of the differential system $(2)$ of the form $(3)$?