| In this thesis, the Rosochatius-type systems (both continuous and discrete) are inves-tigated by virtue of algebraic-geometric tools and generating function technique. These systems are straightened out in the Jacobi variety of the associated hyperelliptic curve. Also the relation between these systems and the soliton equations are revealed. As an ap-plication, the quasi-periodic solution of the KdV equation and Toda equation is obtained in the context of the Rosochatius hierarchy.In the continuous case, we first introduce a hierarchy of Hamiltonian systems re-lated to the Rosochatius-type system (Rosochatius hierarchy for short) based on the deformed Lax matrix. Then we establish the integrability of the Rosochatius hierarchy and straighten out these systems in the Abel-Jacobi coordinates of the associated hyper-elliptic curve. Next, we reveal the relation between these systems and the KdV equation, namely, the first two Rosochatius flows constitute a new integrable decomposition of KdV equation. At last, we calculate the quasi-periodic solution of the KdV equation by the Riemann-theta function and Jacobi inversion.In the discrete case, we propose a method of generating the integrable Rosochatius-type deformations for the integrable symplectic maps. Making use of the method, we obtain the integrable Rosochatius-type deformations of the Toda symplectic map, the Volterra symplectic map and the Ablowitz-Ladik symplectic map. We find their Lax rep-resentations and thus establish their integrabilities in the sense of Liouville with the aid of r-matrix method. Furthermore, we take the Rosochatius-type Toda symplectic map as an illustrative example to show the relation between the Rosochatius-type symplectic maps and the discrete soliton equations. We succeeded in straightening out the Rosochatius-type Toda symplectic map. Based on the new Rosochatius-type integrable decomposition of the Toda equation, we obtain the quasi-periodic solution of the Toda lattice equation.
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