classical mechanics - Integrals of Motion

Landau & Lifshitz write on the first page of chapter 2 of their Mechanics book (p.13)

The number of independent integrals of motion for a closed mechanical system with $s$ degrees of freedom is $2s-1$.

Then they go on

Since the equations of motion for a closed system do not involve time explicitly, the choice of the origin of time is entirely arbitrary, and one of the arbitrary constants in the solution of the equations can always be taken as an additive constant $t_0$ in time. Eliminating $t + t_0$ from the $2s$ functions

$$q_i=q_i(t+t_0,C_1,C_2, \ldots, C_{2s-1}),\qquad \dot{q}_i=\dot{q}_i(t+t_0,C_1,C_2, \ldots, C_{2s-1}),$$ we can express the $2s-1$ arbitrary constants $C_1,C_2, \ldots, C_{2s-1}$ as functions of $q$ and $\dot{q}$ (generalized co-ordinates and velocities) and these functions will be the integrals of the motions.

Could someone elaborate?