\,a_{0},  \eqnum{3.12}$$

$$\partial _{x}B=\kappa \partial _{x}^{2}E+\eta E,  \eqnum{3.13}$$

$$\partial _{x}a_{0}=\chi \partial _{x}B  \eqnum{3.14}$$

The coefficients appearing in these differential equations depend on the components of the polarization operators through the relations

$$\omega =\frac{2\pi }{N}{\it \Pi }_{{\it 0}}\,^{\prime },\quad \alpha =-e^{2}%
{\it \Pi }_{{\it 1}},\quad \tau =e{\it \Pi }_{{\it 0}},\quad \chi =\frac{%
2\pi }{eN},\quad \sigma =-\frac{e^{2}}{\gamma }{\it \Pi }_{{\it 0}},\quad
\eta =-\frac{e^{2}}{\delta }{\it \Pi }_{{\it 1}},$$

$$\gamma =1+e^{2}{\it \Pi }_{{\it 0}}\,^{\prime }-\frac{2\pi }{N}{\it \Pi }_{%
{\it 1}},\quad \delta =1+e^{2}{\it \Pi }_{\,{\it 2}}-\frac{2\pi }{N}{\it \Pi 
}_{{\it 1}},\quad \kappa =\frac{2\pi }{N\delta }{\it \Pi }_{\,{\it 2}}. 
\eqnum{3.15}$$

Distinctive of eq. (3.12) is the presence of the nonzero coefficients $%
\sigma $ and $\tau $. They are related to the Debye screening in the two dimensional anyon thermal ensemble. A characteristic of this 2+1 dimensional model is that the Debye screening disappears at $T=0$, even if the chemical potential is different from zero. Note that $\sigma $ and $\tau $ link the magnetic field to the zero components of the gauge potentials, $A_{0}$ and $%
a_{0}$. As a consequence, these gauge potentials will play a nontrivial role in finding the magnetic field solution of the system.

Field Solutions and Boundary Conditions
---------------------------------------

Using eqs.(3.12)-(3.14) one 