OK, I am back.
In his book "Descendances naturelles des souverains et grands-ducs de Russie de 1762 à 1910" (Paris, 1995), Jacques Ferrand says that a matrimonial union is not to be believed. But the boy, named Alexei, who was born from the love between Grand Duke Alexei Alexandrovitch and Alexandra Vassilievna (who had been a maid-of-honor to Empress Maria Alexandrovna) was titled on March 21, 1884 (I do not know if it is O.S. or N.S.) by Imperial Ukaze "Count Belewsky-Joukovsky".
He married twice. From his first marriage with Princess Maria Petrovna Troubetzkoy (1872-1954), he got four children :
1. Elisabeth (1896-1975). Married twice, two children, two grand-children.
2. Alexandra (1899-?). Married twice, without children.
3. Maria (1901-1996). Married twice, one daughter.
4. Serge (1904-1953). Married, one daughter, three grand-children, two great-grand-children.
Most seem to live in the USA.