5. The first equation of the system defines a pair of parallel lines when $\mathrm{b} \neq 0$ and one line when $b=0$. The second equation defines three lines in general position. Their points of intersection have coordinates $(4 ; 3),(1; 1)$, and $(5; -1)$. Four solutions to the system are possible in the following two cases:
1) The first equation defines a pair of parallel lines (i.e., $\mathrm{b} \neq 0$). These lines are parallel to one of the lines from the second equation of the system and do not pass through their points of intersection.
2) The first equation defines a pair of parallel lines (i.e., $\mathrm{b} \neq 0$), which are not parallel to any of the lines from the second equation of the system, and each of these parallel lines passes through one of the points of intersection of the lines from the second equation.

Consider the first case. Parallelism of lines is checked by the equality of their slopes. The condition for parallelism to the line $2 x-3 y+1=0$ is

$$
-\frac{a}{2}=\frac{2}{3} \Rightarrow a=-\frac{4}{3}
$$

This case is not suitable, since the value of $a$ must be an integer. The condition for parallelism to the line $4 x+y-19=0$ is:

$$
-\frac{a}{2}=-4 \Rightarrow a=8
$$

The value of $b$ is found from the condition of passing through the points of intersection:

$$
b=|8 x+2 y+1|
$$

For the point $(1 ; 1)$, we get $b=11$, for the point $(4 ; 3)$, the value of $b$ is 39, and for the point $(5 ;-1)$, the value of $b$ is also 39. The condition for parallelism to the line $x+2 y-3=0$ is:

$$
-\frac{a}{2}=-\frac{1}{2} \Rightarrow a=1
$$

The value of the parameter is determined by passing through the points of intersection:

$$
b=|x+2 y+1|
$$

For the point $(1 ; 1)$, we get $b=4$, for the point $(4 ; 3)$, the value of $b$ is 11, and for the point $(5 ;-1)$, the value of $b$ is 4. This completes the consideration of the first case.

Consider the second case. Taking into account the previous point, the condition for non-parallelism is

$$
a \neq 8, a \neq 1
$$

The passage through the points of intersection determines the parameter $b$. For the pair of points $(1 ; 1)$ and $(4 ; 3)$, we get

$$
\begin{aligned}
& b=|a+3|=|4 a+7| \\
& {\left[\begin{array}{l}
a+3=4 a+7 \\
a+3=-4 a-7
\end{array}\right.} \\
& {\left[\begin{array}{l}
a=-4 / 3 \\
a=-2
\end{array}\right.}
\end{aligned}
$$

Only the integer value $a=-2$ is suitable, for which $b=1$. For the pair of points $(1 ; 1)$ and $(5 ;-1)$, we have the equation

$$
\begin{aligned}
& b=|a+3|=|5 a-1| \\
& {\left[\begin{array}{l}
a+3=5 a-1 \\
a+3=1-5 a
\end{array}\right.} \\
& {\left[\begin{array}{l}
a=1 \\
a=-1 / 3
\end{array}\right.}
\end{aligned}
$$

Here, neither value is suitable. Finally, for the pair of points $(4 ; 3)$ and $(5 ;-1)$, we get the equation

$$
\begin{aligned}
& b=|4 a+7|=|5 a-1| \\
& {\left[\begin{array}{l}
4 a+7=5 a-1 \\
4 a+7=1-5 a
\end{array}\right.} \\
& {\left[\begin{array}{l}
a=8 \\
a=-2 / 3
\end{array}\right.}
\end{aligned}
$$

Here, neither value is suitable.

Combining all the cases considered, we get the final answer.
Answer: $\left[\begin{array}{l}a=8, b \in N, b \neq 11, b \neq 39 \\ a=1, b \in N, b \neq 4, b \neq 11 \\ a=-2, b=1\end{array}\right.$