Confusion on the cgs, Heavside-Lorentz and SI system I am very confused about three measuring systems: cgs, Heavside-Lorentz (never used) and SI system. I wanted to understand, shortly, the usefulness of the measurement system (HL), because it was not familiar to me.
Into the book the Classical Electrodynamics, Jackson, 3rd edition (Appendix), there is this table 2:
$$\begin{array} {|c|c|}
\hline \textbf{System} & \mathbf{\epsilon_0} & \mathbf{\mu_0} \\ \hline \textbf{Gaussian-cgs} & 1 & 1 \\ \hline \textbf{Heavside-Lorentz (HL)} & 1 & 1 \\ \hline  \end{array}$$
I know that in electostatic in the SI system is $k_e=1/4\pi\epsilon_0$ and in the cgs system is: $k_e=1$ (is it my skill correct?). Therefore I have a different form between the table 2 and my skills. Infact it should be in cgs system $4\pi\epsilon_0=4\pi$ seeing the table 2.
After exists into book also the table 1, where there are, IMHO, $k_1\equiv k_e$, $k_2=k_m$ and who is $\alpha$ and $k_3$?
Table 1:
$$\begin{array} {|c|c|}\hline \textbf{System} & k_1 \\ \hline \textbf{Gaussian-cgs} & 1 \\ \hline \textbf{Heavside-Lorentz (HL)} & \dfrac{1}{4\pi} \\\hline  \end{array}$$
What is the reason of the use of Heavside-Lorentz system?
 A: 
I am very confused about the three measuring systems: cgs, Heavside-Lorentz (first time I saw it written) and SI system. 

The first thing that may be causing confusion is that there is no such thing as the cgs unit system. Cgs is a class of several unit systems that all use the centimeter, gram, and second for their mechanical units, but they differ with respect to their electrical units. Electrostatic units (esu), electromagnetic units (emu), Gaussian units, and Heaviside-Lorentz units (HL) are all distinct cgs unit systems.
The HL system is particularly nice for working with Maxwell's equations. Following the notation in your second table, Maxwell's equations can be written:
$$\nabla \cdot \mathbf E = 4 \pi k_1 \rho$$
$$\nabla \cdot \mathbf B = 0$$
$$\nabla \times \mathbf E = -k_3 \frac{\partial}{\partial t} \mathbf B$$
$$\nabla \times \mathbf B = 4 \pi \alpha k_2 \mathbf J + \alpha \frac{k_2}{k_1}\frac{\partial}{\partial t}\mathbf E$$
So for HL units the above simplifies nicely to
$$\nabla \cdot \mathbf E = \rho$$
$$\nabla \cdot \mathbf B = 0$$
$$c\ \nabla \times \mathbf E = -\frac{\partial}{\partial t} \mathbf B$$
$$c\ \nabla \times \mathbf B =  \mathbf J + \frac{\partial}{\partial t}\mathbf E$$
Thus the main purpose/usefulness of HL units is to work with Maxwell's equations in a more simplified manner. This is similar to working with Newton's second law in SI vs. US customary units. In SI units Newton's 2nd law is $F=ma$, but in US customary units it is $F=kma$ where $k = \frac{1}{32.174}\frac{lb_{f}}{lb_{m}\ ft \ s^2}$. Choosing units consistent with Newton's laws makes using Newton's laws easier. Similarly, choosing units consistent with Maxwell's equations makes using Maxwell's equations easier.

I have seen in the Classical Electrodynamics Jackson 3rd edition (Appendix) table where cgs system is equal to Heavside-Lorentz ... and a table where this system are different.

HL units are different from Gaussian units. In both systems the value of the vacuum permittivity and permeability are set to a dimensionless 1, but the vacuum permittivity and permeability do not by themselves completely define either system. The two tables are not in conflict with each other, they are just discussing different constants. Notice the label at the top of each column, they are listing completely separate characteristics of each system.
