A rabbit initially stands at the position $0$, and repeatedly jumps on the real line. In each jump, the rabbit can jump to any position corresponds to an integer but it cannot stand still. Let $N(a)$ be the number of ways to jump with a total distance of $2019$ and stop at the position $a$. Determine all integers $a$ such that $N(a)$ is odd.
1. **Generating Function Approach:**
   - Consider the generating function for the jumps:
     \[
     T = (x + x^2 + x^3 + \ldots) + (y + y^2 + y^3 + \ldots) = \frac{x}{1-x} + \frac{y}{1-y}
     \]
   - Define the generating function \( F(x, y) \) as:
     \[
     F(x, y) = 1 + T + T^2 + T^3 + \ldots
     \]
   - The coefficient of \( x^a y^b \) in \( F \) represents the number of ways to jump with a total distance of \( a + b \) and arrive at position \( a - b \).

2. **Modulo 2 Evaluation:**
   - To evaluate \( F(x, y) \) modulo 2, consider:
     \[
     G(x, y) = 1 - T + T^2 - T^3 + \ldots
     \]
     so that \( G \equiv F \pmod{2} \).
   - Simplify \( G(x, y) \):
     \[
     G(x, y) = \frac{1}{1 + T} = \frac{1}{1 + \frac{x}{1-x} + \frac{y}{1-y}} = \frac{(1-x)(1-y)}{1 - xy}
     \]
   - Further simplify:
     \[
     G(x, y) = (1 - x - y + xy)(1 + xy + (xy)^2 + (xy)^3 + \ldots)
     \]
   - The odd coefficients in \( G(x, y) \) are of the form \( x^n y^{n+1} \) and \( x^{n+1} y^n \), corresponding to \( N(1) \) and \( N(-1) \).

3. **Combinatorial Approach:**
   - Encode each positive jump by \( + \) and each negative jump by \( - \), separated by \( | \). For example:
     \[
     ++|+++|--- \implies 0 \stackrel{+2}{\longrightarrow} 2 \stackrel{+3}{\longrightarrow} 5 \stackrel{-3}{\longrightarrow} 2
     \]
   - For \( N(a) \), we need \( \frac{2019 + a}{2} \) \( + \)'s and \( \frac{2019 - a}{2} \) \( - \)'s.
   - A sequence is "bad" if there are an odd number of ways to insert \( | \).

4. **Bad Sequences:**
   - A sequence is bad if it has no consecutive \( ++ \) or \( -- \). Thus, the only bad sequences are \( +-+-+ \ldots \) and \( -+-+- \ldots \).
   - These sequences correspond to \( N(1) \) and \( N(-1) \).

Conclusion:
\[
\boxed{\{1, -1\}}
\]