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In this paper, we consider the mean field equation \begin{equation*} \Delta u+e^{u}=\sum _{i=0}^{3}4\pi n_{i}\delta _{\frac {\omega _{i}}{2}}\text { in }E_{\tau }, \end{equation*} where n_{i}\in \mathbb {Z}_{\geq 0}, E_{\tau } is the flat torus with periods \omega _{1}=1, \omega _{2}=\tau and \operatorname {Im}\tau >0. Assuming N=\sum _{i=0}^{3}n_{i} is odd, a non-critical case for the above PDE, we prove: (i) If among \{n_{i}|i=0,1,2,3\} there is only one odd integer, then there is always an even solution. Furthermore, if n_{0} = 0 and n_{3} is odd, then up to SL_{2}(\mathbb {Z}) action, except for finitely many E_{\tau }, there are exactly \frac {n_{3}+1}{2} even solutions. (ii) If there are exactly three odd integers in \{n_{i}|i=0,1,2,3\}, then the equation has no even solutions for any flat torus E_{\tau }. Our second result might suggest the symmetric solution of the above mean field equation does not hold in general. [ABSTRACT FROM AUTHOR]

In this paper, we consider the persistence of completely degenerate lower-dimensional invariant tori of the following reversible system {x = ω + Q(x)z + ε P1(x,z,ε), z + H(z)+ε P2(x,z,ε), where (x,z) = (x,y,u,v) ∈ Tn × Rm × R × R with m ≥ n + 2, Hz) = (0,v2p+1 + yml,uym − 1q)T with y = (y1, ⋅ ⋅ ⋅, ym−1,ym), p,q ≥ 0, l > 0 are integers, the involution G is (x,y,u,v) → (−x,y,−u,v), Q(x) is a n × m matrix function, ω is a Diophantine frequency, ε is a small positive parameter and ε P1 ε P2 are analytic perturbation terms. By the Kolmogorov-Arnold-Moser method, we prove that for sufficiently small ε the above reversible system admits lower-dimensional invariant tori with prescribed frequency ω if average of a part of Q(x) is non-singular. This should be the first persistence result of lower-dimensional invariant tori in completely degenerate reversible systems. [ABSTRACT FROM AUTHOR]