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Two Flows Kowalevski Top as the Full Genus Two Jacobi's Inversion Problem and Sp(4,$\mathbb{R}$) Lie Group Structure

By using the first and the second flows of the Kowalevski top, we can make the Kowalevski top into the two flows Kowalevski top, which has two time variales. Then we show that equations of the two flows Kowalevski top become those of the full genus two Jacobi inversion problem. In addition to the Lax pair for the first flow, we costruct Lax pair for the second flow. Using the first and the second flows, we show that the Lie group structure of these two Lax pairs is Sp(4,$\mathbb{R}$) $\cong$ SO(3,2). Through the two flows Kowalevski top, we can conclude that the Lie group structure of the genus two hyperelliptic function is Sp(4,$\mathbb{R}$) $\cong$ SO(3,2).