1. (14 points) Let the line $l: y=k x+m(k, m \in$ Z) intersect the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{12}=1$ at two distinct points $A, B$, and intersect the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{12}=1$ at two distinct points $C, D$. Question: Does there exist a line $l$ such that the vector $\overrightarrow{A C}+\overrightarrow{B D}=\mathbf{0}$? If it exists, indicate how many such lines there are; if not, explain the reason.
Given the system of equations:
$$
\left\{\begin{array}{l}
y=k x+m, \\
\frac{x^{2}}{16}+\frac{y^{2}}{12}=1,
\end{array}\right.
$$
eliminating \( y \) and simplifying, we get:
$$
\left(3+4 k^{2}\right) x^{2}+8 k m x+4 m^{2}-48=0.
$$

Let \( A\left(x_{1}, y_{1}\right) \) and \( B\left(x_{2}, y_{2}\right) \). Then:
$$
\begin{array}{l}
x_{1}+x_{2}=-\frac{8 k m}{3+4 k^{2}}. \\
\Delta_{1}=(8 k m)^{2}-4\left(3+4 k^{2}\right)\left(4 m^{2}-48\right)>0.
\end{array}
$$

From the system:
$$
\left\{\begin{array}{l}
y=k x+m, \\
\frac{x^{2}}{4}-\frac{y^{2}}{12}=1,
\end{array}\right.
$$
eliminating \( y \) and simplifying, we get:
$$
\left(3-k^{2}\right) x^{2}-2 k m x-m^{2}-12=0.
$$

Let \( C\left(x_{3}, y_{3}\right) \) and \( D\left(x_{4}, y_{4}\right) \). Then:
$$
x_{3}+x_{4}=\frac{2 k m}{3-k^{2}}.
$$
$$
\Delta_{2}=(-2 k m)^{2}+4\left(3-k^{2}\right)\left(m^{2}+12\right)>0.
$$

Since \( \overrightarrow{A C}+\overrightarrow{B D}=0 \), we have:
$$
\left(x_{4}-x_{2}\right)+\left(x_{3}-x_{1}\right)=0.
$$

Thus, \( \left(y_{4}-y_{2}\right)+\left(y_{3}-y_{1}\right)=0 \).
From \( x_{1}+x_{2}=x_{3}+x_{4} \), we get:
$$
-\frac{8 k m}{3+4 k^{2}}=\frac{2 k m}{3-k^{2}}.
$$

Therefore, \( 2 k m=0 \) or \( -\frac{4}{3+4 k^{2}}=\frac{1}{3-k^{2}} \).
Solving the above, we get \( k=0 \) or \( m=0 \).
When \( k=0 \), from equations (1) and (2), we get:
$$
-2 \sqrt{3}<m<2 \sqrt{3}.
$$

Since \( m \) is an integer, the values of \( m \) are:
$$
-3,-2,-1,0,1,2,3.
$$

When \( m=0 \), from equations (1) and (2), we get \( -\sqrt{3}<k<\sqrt{3} \). Since \( k \) is an integer, the values of \( k \) are:
$$
-1,0,1.
$$

Thus, there are 9 lines that satisfy the conditions.