A special case of Neumann's system and the Kowalewski-Chaplygin-Goryachev top

The Kowalewski top: A new Lax representation

“…It obeys the FCR (4) with the rational R-matrix of XXX type. For the Toda lattices associated with the Lie algebras of the series B n , C n and D n a similar correspondence has been introduced in [15]. Such relations can be helpful for the non-three diagonal matrices.…”

“…We will start with the 4 × 4 Lax representation for the KT given by Reyman and Semenov-Tian-Shansky [8], which obeys (11).…”

“…Since for the quantization of the KT we prefer to construct a new monodromy matrix for the KT in the 2 × 2 auxiliary space. Building such matrix we will use a known matrix in larger auxiliary space, which satisfies a linear R-matrix algebra (11).…”

“…For this purpose we will use often the geometrical [10] and algebraic [11] connections of the tops on e(3) and the Toda lattices. As a settlement, we oblige to introduce the additive deformations of the basic algebraic relations (4-9) and we obtain a Lax triad for the KT…”

Yang-Baxter equation and reflection equations in integrable models

“…The realization we are looking for defined by the generators J k , x k , k = 1, 2, 3 of the Lie algebra e(3), provided that l = 3 k=1 x k J k = 0 [46] y 0 = 1 y 4 = ibx 1 y 1 = −(J 2 1 + J 2 2 + 1 4 + 2α/x 2 3 ) y 5 = ibx 2 y 2 = −2J 3 y 6 = − 1 2 ib{x 3 , J 1 } y 3 = b 2 x 2 3…”