The Half-period Addition Formulae for Genus Two Hyperelliptic $\mathrm{\wp }$ Functions and the Sp(4,$\mathbb{R}$) Lie Group Structure

In the previous study, by using the two-flows Kowalevski top, we have demonstrated that the genus two hyperelliptic functions provide the Sp(4,$\mathbb{R}$)/$Z_2$ $\cong$ SO(3,2) Lie algebra structure. In this study, by directly using the differential equations of the genus two hyperelliptic $\mathrm{\wp }$ functions instead of using integrable models, we demonstrate that the half-period addition formula for the genus two hyperelliptic functions provides the order two Sp(4,$\mathbb{R}$) Lie group structure.