Describe the bug

The response template value is invalid and results in a 500. The 500 response error is caused by an error thrown trying to transform the body because the velocity template language is invalid.

The issue is here: https://github.com/aws/aws-cdk/blob/main/packages/aws-cdk-lib/aws-apigateway/lib/integrations/stepfunctions.ts#L217

It should be this instead:

{
  /**
   * Match all '5XX' HTTP Errors
   */
  selectionPattern: '5\\d{2}',
  statusCode: '500',
  responseTemplates: {
    'application/json': `{
      "error": "$input.path('$.error')"
    }`,
  },
},

Regression Issue

• Select this option if this issue appears to be a regression.

Last Known Working CDK Library Version

No response

Expected Behavior

Transform the body of the 500 integration response without erroring on the transformation.

Current Behavior

API Gateway throws an error on the transformation of the response because the VTL syntax is wrong.

Reproduction Steps

Deploy this construct and trigger a 500 response from the integration sdk request. Or use this bad VTL template in a 200 you'll see API Gateway error trying to transform the payload.

Possible Solution

The 400 template is correct, so re-use it but with the error value.

{
  /**
   * Match all '5XX' HTTP Errors
   */
  selectionPattern: '5\\d{2}',
  statusCode: '500',
  responseTemplates: {
    'application/json': `{
      "error": "$input.path('$.error')"
    }`,
  },
},

Additional Information/Context

The current implementation is not valid JSON, it's just a string but it's expecting a JSON response.

AWS CDK Library version (aws-cdk-lib)

2.202.0

AWS CDK CLI version

4.0.0

Node.js Version

20.18.1

OS

osx

Language

TypeScript

Language Version

No response

Other information

No response