Hi, Unfortunately, there is nothing you can do.
As shown below, API page type cannot be extended by creating a page extension object. Instead, you must create a new API by adding a page object.

So, you will need to follow the link provided by Inge and create a new API page to do so.
Here is the standard itemLedgerEntries page code that you use, hope it helps you.
page 30069 "APIV2 - Item Ledger Entries"
{
APIVersion = 'v2.0';
EntityCaption = 'Item Ledger Entry';
EntitySetCaption = 'Item Ledger Entries';
DelayedInsert = true;
DeleteAllowed = false;
Editable = false;
EntityName = 'itemLedgerEntry';
EntitySetName = 'itemLedgerEntries';
InsertAllowed = false;
ModifyAllowed = false;
PageType = API;
SourceTable = "Item Ledger Entry";
Extensible = false;
ODataKeyFields = SystemId;
layout
{
area(content)
{
repeater(Group)
{
field(id; SystemId)
{
Caption = 'Id';
Editable = false;
}
field(entryNumber; "Entry No.")
{
Caption = 'Entry No.';
Editable = false;
}
field(itemNumber; "Item No.")
{
Caption = 'Item No.';
}
field(postingDate; "Posting Date")
{
Caption = 'Posting Date';
}
field(entryType; "Entry Type")
{
Caption = 'Entry Type';
}
field(sourceNumber; "Source No.")
{
Caption = 'Source No.';
}
field(sourceType; "Source Type")
{
Caption = 'Source Type';
}
field(documentNumber; "Document No.")
{
Caption = 'Document No.';
}
field(documentType; "Document Type")
{
Caption = 'Document Type';
}
field(description; Description)
{
Caption = 'Description';
}
field(quantity; Quantity)
{
Caption = 'Quantity';
}
field(salesAmountActual; "Sales Amount (Actual)")
{
Caption = 'Sales Amount (Actual)';
}
field(costAmountActual; "Cost Amount (Actual)")
{
Caption = 'Cost Amount (Actual)';
}
field(lastModifiedDateTime; SystemModifiedAt)
{
Caption = 'Last Modified Date';
}
}
}
}
actions
{
}
}
Thanks.
ZHU