Periodic solution to integral

I've been trying to find a vector function where $$A_x(x,y)$$ and $$A_y(x,y)$$ are periodic functions with period $$\ell_q$$, of the form $$A(x,y)=(By+A_x(x,y),A_y(x,y))$$ such that the discretised version of $$F(x_1,x_2):=(x_1,x_2)\cdot\int_0^1A\left(q_1-sx_1,q_2-sx_2\right)ds,$$ i.e., $$x_1=\frac{\ell_q m_1}{N}$$, $$x_2=\frac{\ell_q m_2}{N}$$, $$q_1=\frac{\ell_q l_1}{N}$$ $$q_2=\frac{\ell_q l_2}{N}$$, is $$2Np$$ periodic in the $$m$$ argument, i.e.,

$$F(m_1+2Np_1, m_2+2Np_2)=F(m_1,m_2)$$ and when $$p_1,p_2\in\mathbb{Z}$$

The linear term in $$2Np$$ periodic when $$B=2\pi n$$ when $$n\in\mathbb{Z}$$, but I can see to find a condition for the rest of the vector function.

Does anybody know a way that would help?

Thanks.