A grasshopper is sitting at an integer point in the Euclidean plane. Each second it jumps to another integer point in such a way that the jump vector is constant. A hunter that knows neither the starting point of the grasshopper nor the jump vector (but knows that the jump vector for each second is constant) wants to catch the grasshopper. Each second the hunter can choose one integer point in the plane and, if the grasshopper is there, he catches it. Can the hunter always catch the grasshopper in a finite amount of time?


1. **Understanding the Problem:**
   - The grasshopper starts at an integer point \((x_0, y_0)\) in the Euclidean plane.
   - Each second, it jumps to another integer point with a constant jump vector \((x_1, y_1)\).
   - The hunter does not know the starting point \((x_0, y_0)\) or the jump vector \((x_1, y_1)\), but knows that the jump vector is constant.
   - The hunter can choose one integer point in the plane each second to check if the grasshopper is there.
   - The goal is to determine if the hunter can always catch the grasshopper in a finite amount of time.

2. **Formulating the Grasshopper's Position:**
   - At time \(t\), the position of the grasshopper is given by:
     \[
     (x_0 + x_1 t, y_0 + y_1 t)
     \]
   - Here, \((x_0, y_0)\) is the initial position and \((x_1, y_1)\) is the constant jump vector.

3. **Reduction to an Algorithmic Problem:**
   - The problem reduces to finding an algorithm that can identify all possible ordered quadruplets \((x_0, x_1, y_0, y_1) \in \mathbb{Z}^4\).
   - The hunter needs to check all possible integer points in the plane in such a way that every possible trajectory of the grasshopper is eventually checked.

4. **Spiral Search Strategy:**
   - One effective strategy is to use a spiral search pattern starting from the origin \((0, 0)\).
   - This method ensures that every integer point in the plane is eventually checked.
   - The spiral search can be visualized as iterating through all points in a manner similar to enumerating rational numbers.

5. **Ensuring Finite Time Capture:**
   - Since the hunter checks one point per second and the spiral search covers all integer points, every possible initial position and jump vector will be checked in finite time.
   - Therefore, the hunter will eventually catch the grasshopper.

Conclusion:
\[
\boxed{\text{True}}
\]