Begin by setting up your Kafka environment. Install Apache Kafka on your system by downloading the latest version from the official Kafka website. Ensure you also have Apache Zookeeper installed, as Kafka depends on it for managing the Kafka brokers. Start Zookeeper and then start Kafka using the provided scripts (`zookeeper-server-start.sh` and `kafka-server-start.sh`).

Create a Kafka topic where the API data will be published. Use the Kafka command-line tool to create a topic, specifying the desired number of partitions and replication factor. Example command:
```bash
kafka-topics.sh --create --topic my-api-data --bootstrap-server localhost:9092 --partitions 1 --replication-factor 1
```

Write a script to fetch data from the public API. Choose a programming language like Python, Node.js, or Java, and use its standard HTTP library to make GET requests to the API. Ensure the script can handle pagination and rate limits if applicable. Example in Python using `requests` library:
```python
import requests

def fetch_data(api_url):
response = requests.get(api_url)
if response.status_code == 200:
return response.json()
else:
response.raise_for_status()
```

In the same script, set up a Kafka producer to send the fetched data to the Kafka topic. Use a Kafka client library for your programming language of choice (e.g., `kafka-python` for Python). Configure the producer with the necessary Kafka broker information. Example in Python:
```python
from kafka import KafkaProducer
import json

producer = KafkaProducer(
bootstrap_servers='localhost:9092',
value_serializer=lambda v: json.dumps(v).encode('utf-8')
)

def send_to_kafka(data):
producer.send('my-api-data', value=data)
producer.flush()
```

Integrate the data fetching and Kafka sending logic in your script. This involves calling the fetch function, processing the response if necessary, and passing it to the Kafka producer to send to the topic. Ensure proper exception handling to manage any errors in data fetching or sending.

If continuous data transfer is needed, schedule the script to run at regular intervals. Use a cron job on Unix-based systems or Task Scheduler on Windows to automate the script execution. For example, a cron entry to run the script every hour might look like:
```bash
0 /usr/bin/python3 /path/to/your/script.py
```

Implement logging within your script to monitor its operation and diagnose any issues. Log API responses, data sent to Kafka, and any errors encountered. Use Kafka's management tools to monitor the health of your Kafka cluster and topic throughput. Adjust the script based on log insights and performance needs.

By following these steps, you can efficiently move data from a public API to a Kafka topic without relying on third-party connectors or integrations.