Let \(\text{MK}(c;\mu_i)=\text{sup}\left\{\int_{\Omega^N}c(x_i)\,d\pi;\;{\mathcal P}(\Omega^N)\right\}\), where \(i=1,\dots,N\), \(\mu_i\) are Borel probability measures on a domain \(\Omega\) of \(\mathbb R^d\), and \(c:\Omega^N\to\mathbb R\cup\{-\infty\}\) is a bounded Borel cost function. The multi-marginal version of the Monge-Kantorovich problem is to maximize \(\text{MK}\) among all probability measures \(\pi\) on \(\Omega^N\) with \(\text{proj}_i\pi=\mu_i\). If \(\pi\in{\mathcal P}_{\text{sym}}(\Omega^N,\mu)\), where \({\mathcal P}_{\text{sym}}(\Omega^N,\mu)\) is the set of Radon probability measures on \(\Omega^N\) which are invariant under the cyclic permutation \(\sigma(x_1,\dots,x_N)=(x_2,\dots,x_N,x_1)\), then \(\text{MK}_{\text{sym}}(c;\mu)\) is the symmetric version of \(\text{MK}\). A map \(u:\Omega\to\mathbb R^d\) is said to be \(N\)-cyclically monotone, if \(\sum^{N}_{i=1}\langle u(x_i),x_i-x_{i+1}\rangle\geq 0\) for every cycle \(x_1,\dots,x_N,x_{N+1}=x_1\) of points in \(\Omega\). A family of vector fields \(u_1,\dots,u_{N-1}\) from \(\Omega\) to \(\mathbb R^d\) is said to be jointly \(N\)-monotone if \(\sum^{N}_{i=1}\sum^{N-1}_{l=1}\langle u_l(x_i),x_i-x_{i+l}\rangle\geq 0\) for every cycle \(x_1,\dots,x_{2N-1}\) of points in \(\Omega\) such that \(x_{N+l}=x_l\) for \(1\leq l\leq N-1\). If there exists an \(N\)-antisymmetric Hamiltonian \(H\) that is concave in the first variable and convex in the last \((N-1)\) variables such that \((u_1(x),\dots,u_N(x))=\nabla_{2,\dots,N}H(x,Sx,\dots,S^{N-1}x)\), then it is called a polar decomposition for \((u_1(x),\dots,u_N(x))\).
In this paper, the authors consider symmetric Monge-Kantorovich problems and polar decompositions of vector fields. The main goal of the paper is to investigate what happens when \(u_1(x),\dots,u_N(x)\) are arbitrary bounded vector fields, and in particular whether \(\text{MK}_{\text{sym}}(c;\mu)\) is attained at some \(S\in{\mathcal S}_N(\Omega,\mu)\) when \(c\) is the cost given by \(c(x)=\langle u_1(x_1),x_2\rangle+\dots+\langle u_{N-1}(x_1),x_{x_N}\rangle\), where \({\mathcal S}_N(\Omega,\mu)\) is the set of \(\mu\)-preserving transformations on \(\Omega\). The problem of the existence of a polar decomposition is studied.