Which physical system do these dynamic equations represent? \begin{align*} \dot{x}_1&=bx_1+kx_2+x_3 \\\ \dot{x}_2&=x_1 \\\ \dot{x}_3&=\alpha(u-x_2)-\beta x_3. \end{align*} Can someone help me identify what is the physical representation of the above dynamic system and what would the coefficient correspond to? Looks a lot like van der Pol oscillator, or some other type of oscillator.

Since the system is LTI it won't act as a van der Pol oscillator, since that requires nonlinear dynamics. The number of parameters would allow you to place the poles/eigenvalues of the system anywhere you want, so this general third order system wouldn't be limited to a any subset of third order dynamics.

That being said, I do not know any good examples of truly third order systems. Namely many mechanical systems are of order two (or an integer multiple of two). Maybe you can interpret the system as two masses connected by a damper and a spring, which is sliding as a whole over a viscus horizontal surface. This would technically be a fourth order system with one pole/eigenvalue at zero. But by choosing the states as (a linear combination of) the relative-distance and -velocity between the two masses and the velocity of the center of mass, the dynamics can also be described as third order.