Rotationally symmetric symphonic maps

We consider a functional of pullbacks of metrics on the space of maps f between Riemannian manifolds. Harmonic maps are stationary points of the energy functional E(f) which is an integral of the trace of the pullback of the metric of the target manifold by f. Our functional Esym(f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\mathrm{sym} (f)$$\end{document} is an integral of the norm of the pullback. Stationary maps for Esym(f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\mathrm{sym}(f)$$\end{document} are called as symphonic maps (Kawai in Nonlinear Anal. 74: 2284-2295, 2011), (Kawai in Differ. Geom. Appl. 44: 161-177, 2016), (Misawa in Nonlinear Anal. 75: 5971-5974, 2012), (Misawa in Calc. Var. Part. Differ. Equ. 55: 1-20, 2016), (Misawa in Adv. Differ. Equ. 23: 693-724, 2018), (Misawa in Equ. Appl. 2: 1-20, 2021), (Misawa in Adv. Geom. 22: 23-31, 2022), (Nakauchi in Nonlinear Anal. 108: 87-98, 2014) and (Nakauchi in Ricerche di Matematica 60: 219-235, 2011). In this paper, we are concerned with rotationally symmetric maps. We prove that any rotationally symmetric map between 4-dimensional model spaces is a symphonic map if and only if it is a conformal map.