(where $v^{A}_{1}$ and $v^{A}_{2}$ are eigenvalues of $u^{A}_{1}$ and $u^{A}_{2}$). The solutions include two arbitrary complex functions with $Z_4$ symmetry. The kernels of the integrations are closed analytic functions of $u_{1}$ and $u_{2}$. In the simplest case, when $A$ is an even number, we reobtain the Boca’s classic result [@16] and obtain a new result when $A$ is an odd number. Moreover the above construction is also applicable to the integral $T^2/Z_N (N=3,6)$ cases.

This paper is organized as following: In Section 2, we introduce operators on the noncommutative orbifold $T^2/Z_N$. In Section 3, we introduce the $%
|k,q>$ representation and provide the matrix element relation for the projectors and deduce the relation between the eigen value functions of coefficients and the matrix elements of operators in the $|k,q>$ representation. In Section 4, we study the general projectors with minimal trace when the eigen value functions of coefficients are continuous. In Section 5, we present two kinds of explicit expressions for the projectors with elliptical functions as kernel.

Noncommutative Orbifold $T^2/Z_N$
=================================

In this section, we introduce operators on the noncommutative orbifold $%
T^{2}/Z_{N}$. First we introduce two hermitian operators $\hat{y_{1}}$ and $%
\hat{y_{2}}$, which satisfy the following commutation relation: 