Entire functions in weighted $L_2$ and zero modes of the Pauli operator with non-signdefinite magnetic field

For a real non-signdefinite function $B(z)$, $z\in \C$, we investigate the dimension of the space of entire analytical functions square integrable with weight $e^{\pm 2F}$, where the function $F(z)=F(x_1,x_2)$ satisfies the Poisson equation $\D F=B$. The answer is known for the function $B$ with constant sign. We discuss some classes of non-signdefinite positively homogeneous functions $B$, where both infinite and zero dimension may occur. In the former case we present a method of constructing entire functions with prescribed behavior at infinity in different directions. The topic is closely related with the question of the dimension of the zero energy subspace (zero modes) for the Pauli operator.