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For a real non-signdefinite function $B(z)$, $z\in \C$, we investigate thedimension of the space of entire analytical functions square integrable withweight $e^{\pm 2F}$, where the function $F(z)=F(x_1,x_2)$ satisfies the Poissonequation $\D F=B$. 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 wepresent a method of constructing entire functions with prescribed behavior atinfinity in different directions. The topic is closely related with thequestion of the dimension of the zero energy subspace (zero modes) for thePauli operator.

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