A counterexample to analytic hypoellipticity of \(\overline\partial_b\) for a real analytic \(CR\) manifold of finite type given by Christ and Geller is as follows: The Szégö kernel of \(M_m(m=2,3, \dots)\) fails to be real analytic off of the diagonal. As a corollary a three dimensional \(CR\) manifold, \(M_m=\{I_mz_2 =[R_ez_1]^{2m}\}\) \((m=2,3, \dots)\), \(\overline \partial_b\) fails to be relatively analytic hypoelliptic. The present paper considers the hypersurface \(M_m= \{I_mz_2=P(z_1) \subset\mathbb{C}^2\), where \(P:\mathbb{C} \mapsto\mathbb{R}\) is a subharmonic, nonharmonic polynomial\}. Such a surface is pseudoconvex and of finite type. A nonvanishing, antiholomorphic tangent vector field is \({\partial\over\partial \overline z_1}-2i ({\partial P\over\partial \overline z_1}) {\partial\over\partial \overline z_1}\). The coordinates for the surface are \(C\times\mathbb{R}\ni (z=x+iy,t) \mapsto (z,t+iP(z))\). The vector field pulls back to \(\overline\partial_b= {\partial\over\partial \overline z} -i({\partial P\over\partial \overline z_1}) {\partial\over\partial t}\). The formal adjoint is denoted \(\overline\partial^*_b\). When \(M=M_m\), Christ constructed singular solutions for \(\overline\partial_bu=0\) \((u=\overline\partial^*_b v,\;v\in L^2)\) by Fourier transforms. Letting \(s(z,t)\) be the Szégö kernel of \(M\) and if \(M=M_m\) \((m=2,3, \dots)\) the author gives a representation for the counterexample to be \(K(z,t)= S((x,t);\;(0,0))\) as \(K(z,t)= c\int^\infty_0e^{-p} H(z,t;p)dp\) whenever \(|\arg z\pm {\pi\over 2} |< {1 \over 2m-1} \cdot {\pi\over 2}\) and \(H(z,t;p)= \sum^\infty_{j=1} c_jS_j^{1\over m} (z,t)p^{f(j)}\) for some \(f(j)=j+j_0+ O({1\over j})\) as \(j\to\infty\). Further refined results are also established.