SQL Server 2014

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CREATE TABLE #temptable(
  [SourceSystem] nchar(3),
  [RunDate] datetime,
  [ResubmitCount] int,
  [FromDate] date,
  [ToDate] date
);
INSERT INTO #temptable VALUES
  (N'ILG', N'2021-07-28T15:35:23.207', 0, N'2021-06-01T00:00:00', N'2021-06-01T00:00:00'),
  (N'ILG', N'2021-07-28T15:35:23.707', 0, N'2021-06-05T00:00:00', N'2021-06-06T00:00:00'),
  (N'AAP', N'2021-07-28T15:35:23.833', 0, N'2021-06-01T00:00:00', N'2021-06-02T00:00:00'), 
  (N'AAP', N'2021-07-28T15:35:23.833', 0, N'2021-06-04T00:00:00', N'2021-06-04T00:00:00'), 
  (N'ZZP', N'2021-07-28T15:35:23.897', 0, N'2021-06-05T00:00:00', N'2021-06-05T00:00:00');
5 rows affected

WITH
  a AS (
    SELECT
      SourceSystem, FromDate, ToDate,
      LEAD(FromDate) OVER(
        PARTITION BY SourceSystem
        ORDER BY RunDate
      ) AS NextDate
    FROM #temptable
  )
SELECT
  SourceSystem,
  DATEADD(DAY, 1, ToDate) AS GapBeg,
  DATEADD(DAY, -1, NextDate) AS GapFin
FROM a
WHERE
  NextDate IS NOT NULL AND
  DATEADD(DAY, -2, NextDate) >= ToDate;


SourceSystem
GapBeg
GapFin


AAP
2021-06-03
2021-06-03

ILG
2021-06-02
2021-06-04