The authors consider finite-dimensional Hamiltonian systems which can be naturally derived from the so-called Betchov-da Rios equation (also called ``localized induction equation''), \[ \frac{\partial\gamma}{\partial t}= \Biggl[\frac{\partial\gamma} {\partial s},\;\frac{\partial^ 2 \gamma} {\partial s^ 2}\Biggr],\tag{1} \] which is a known model equation for thin vortex tubes in an incompressible inviscid three-dimensional (3D) flow. It is known that equation (1) is integrable and admits exact soliton solutions. As a typical integrable system, it possesses an infinite set of integrals of motion. In the present work, the third, fourth, and fifth integrals of motion are employed to construct integrable finite-dimensional systems. Using the natural geometric origin of equation (1), the authors consider lines in 3D with a finite number of free parameters (they actually take, respectively, \(n= 6,7\), or 8 degrees of freedom). They insert the corresponding representations into the above-mentioned conserved integrals, and, using them as Hamiltonians, derive the Hamiltonian systems with \(n\) degrees of freedom. To prove their Liouville integrability, they demonstrate that the number of integrals of motion of the systems is exactly equal \(n\). In each case, five conserved quantities exist automatically due to the geometric origin of the systems, and existence of additional integrals of motion is proved for each system using special properties of the corresponding Hamiltonians.