We solve the continuation problem posed by Maz'ya for the fundamental solution of the non-local parabolic Cauchy problem on the unit circle generated by\[(Au)(x)=\int_C\frac{u(x)-u(y)}{|x-y|}\,|dy|.\]The main point is to distinguish two objects which are often conflated: the distribution-valued fundamental solution and its pointwise diagonal value. We realize $A$ as the non-negative self-adjoint Fourier multiplier on $L^2(\T)$ with eigenvalues\[\lambda_0=0,\qquad \lambda_n=4H_{2n}-2H_n\quad(n\geq1).\]The fundamental solution\[K_z(\theta)=\frac{1}{2\pi}\sum_{n\in\Z}\exp(-z\lambda_{|n|})\e^{in\theta}\]extends to an entire $\D'(\T)$-valued function of $z\in\C$. For $\Re z>1/2$ the diagonal value is an ordinary value of an absolutely convergent Fourier series and is given by\[K_z(0)=\frac1{2\pi}\bigl(1+2Z_A(z)\bigr),\qquadZ_A(z)=\sum_{n=1}^{\infty}\exp[-z(4H_{2n}-2H_n)].\]The scalar function $Z_A$ has a meromorphic continuation to the whole plane, with at most simple poles at $z=1/2-m$, $m=0,1,2,\ldots$. If $z_m=1/2-m$, then\[\Res_{z=z_m}Z_A(z)=\frac12\e^{-c_0z_m}p_m(z_m),\qquad c_0=4\log2+2\gamma,\]where the polynomials $p_m$ are given explicitly by finite Bell-polynomial formulas. In particular,\[\Res_{z=1/2}Z_A(z)=\frac{\e^{-\gamma}}{8},\qquad\Res_{z=-1/2}Z_A(z)=\frac{\e^{\gamma}}{12},\qquad\Res_{z=-3/2}Z_A(z)=-\frac{9\e^{3\gamma}}{20}.\]For $\theta\not\equiv0\pmod{2\pi}$, the pointwise kernel $K_z(\theta)$ is entire in $z$. Thus the answer to Maz'ya's question is: the fundamental solution is entire in the distributional sense, while its diagonal scalar trace is meromorphic with the pole structure above.