The paper under review studies fixed points of the weighted Berezin transform on the polydisk \(\mathbb{D}^n\) for \(n\geq 2\). Many authors have studied this problem for the case \(n=1\) and for different domains in higher dimensions. For \(c>-1\), let \(\nu_c\) be a measure on \(\mathbb{C}\) defined by \(d\nu_c(z)=(c+1)(1-|z|^2)^c\,dA(z)\) so that \(\nu_c(\mathbb{D})=1\). For \(f\in L^1(\mathbb{D},\nu_c)\) and \(z\in\mathbb{D}\), the weighted Berezin transform of \(f\) is defined by \[ (B_cf)(z)=\int_\mathbb{D}(f\circ\varphi_z)\,d\nu_c, \quad\text{where }\varphi_z\in\Aut(\mathbb{D}). \] If \(f\in L^1(\mathbb{D},\nu_c)\) is harmonic, then it is easy to see that \(B_cf=f\). It follows from the work of Furstenberg that the converse is true for \(f\in L^{\infty}(\mathbb{D})\). \textit{P. Ahern, M. Flores} and \textit{W. Rudin} [J. Funct. Anal. 111, No. 2, 380--397 (1993; Zbl 0771.32006)] proved that, for the unweighted Berezin transform on the unit ball \(B\) in \(\mathbb{C}^n\) and \(f\in L^1(B,d\nu)\), \(Bf=f\) implies that \(f\) is \(\mathcal{M}\)-harmonic if and only if \(n\leq 11\). For simplicity, we assume \(n=2\). For \(c_1,c_2>-1\) and \(f\in L^1(\mathbb{D}^2,\nu_{c_1}\times\nu_{c_2})\), the weighted Berezin transform on \(\mathbb{D}^2\) is defined by \[ (B_{c_1,c_2}f)(z,w)=\int_\mathbb{D}\int_\mathbb{D} f(\varphi_z(x),\varphi_w(y))\,d\nu_{c_1}(x)\,d\nu_{c_2}(y). \] Instead of harmonic functions, it is appropriate to study 2-harmonic functions on the polydisk, namely, functions that are harmonic in each variable. If \(f\in C^2(\mathbb{D}^2)\) is 2-harmonic, then again, \(B_{c_1,c_2}f=f\), and the converse is true for bounded functions on \(\mathbb{D}^2\). The main result of this paper shows that, for every \(p\in[1,\infty)\) and \(c_1,c_2>-1\), there exists \(f\in L^p(\mathbb{D}^2,\nu_{c_1} \times \nu_{c_2})\) which is not 2-harmonic but it satisfies \(B_{c_1,c_2}f=f\). The proof uses some ideas from the paper of Ahern, Flores and Rudin [loc. cit.]. The author finally shows that \(B_{c_1,c_2}f=f\) does imply that \(f\) is 2-harmonic provided the radialization of \(f\circ \psi\) belongs to \(L^{\infty}(\mathbb{D}^2)\) for every \(\psi\in\Aut(\mathbb{D})\), thus generalizing Lemma~2 of \textit{S. Axler} and \textit{Ž. Čučković} [Integral Equations Operator Theory 14, No. 1, 1--12 (1991; Zbl 0733.47027)].