Begin by manually exporting the data you require from Wrike. Wrike allows you to export data in CSV or Excel format. Navigate to the specific report or data set in Wrike, and use the export feature to download the data to your local machine. Ensure that all necessary data fields are included in the export.

Set up your local environment to handle data transformation. Install necessary tools such as Python and Apache Spark if they are not already available. These tools will help in transforming the data into a format compatible with Apache Iceberg.

Use a script, preferably in Python with the Pandas and PySpark libraries, to read the exported CSV/Excel file and convert it into Parquet format. Parquet is a columnar storage file format that is highly efficient for analytical queries and is supported by Apache Iceberg. This step involves parsing the CSV/Excel and writing it as a Parquet file.

Example in Python:
```python
import pandas as pd
from pyspark.sql import SparkSession

# Initialize Spark
spark = SparkSession.builder.appName("WrikeToIceberg").getOrCreate()

# Load data
df = pd.read_csv("exported_wrike_data.csv")

# Convert to Spark DataFrame
spark_df = spark.createDataFrame(df)

# Write to Parquet
spark_df.write.parquet("wrike_data.parquet")
```

Ensure that Apache Iceberg is configured properly in your environment. Iceberg can be run on top of existing Hadoop or Spark setups. Install Iceberg by following the official documentation and configure it to work with your data storage system (HDFS, S3, etc.).

Use Apache Spark or any other compatible engine to create an Iceberg table. This involves defining the schema that matches your Parquet data and setting up the table in your Iceberg catalog. You can use Spark SQL to achieve this.

Example with Spark SQL:
```sql
CREATE TABLE iceberg_table (
id STRING,
name STRING,
status STRING,
...
) USING iceberg;
```

With the Iceberg table created, load the Parquet data into it. Use Apache Spark to read the Parquet file and write it into the Iceberg table. This process ensures your data is now part of the Iceberg data lake.

Example in PySpark:
```python
# Read Parquet data
parquet_df = spark.read.parquet("wrike_data.parquet")

# Write to Iceberg table
parquet_df.writeTo("iceberg_table").append()
```

Once the data is loaded, perform integrity checks to ensure that the data in the Iceberg table matches the original data exported from Wrike. Use queries to compare counts, check data types, and verify that all records have been accurately transferred. This step ensures the reliability of your data migration process.

By following these steps, you can successfully move data from Wrike to Apache Iceberg without relying on third-party connectors or integrations, while maintaining data integrity and compatibility with Apache Iceberg's requirements.