Finite-Gap Solutions of the Pohlmeyer--Lund--Regge Equation and the Associated Curve Evolution

We develop a finite-gap construction for the Pohlmeyer--Lund--Regge (PLR) equation and the associated Lund--Regge curve evolution. From the hyperelliptic spectral data we build a Baker--Akhiezer function and an $\mathrm {SU}(2)$-frame, yielding an explicit theta-quotient formula for the PLR solution. We then derive criteria of the Lund-Regge curve: under natural quasi-periodicity assumptions, $s$-closure and $t$-periodicity are each equivalent to a critical-point condition for the corresponding quasimomentum differential together with a phase quantization at the reconstruction point. This provides a PLR analogue of the closure mechanism of Calini--Ivey.