Is it permissible for a sister to marry someone "by Fatihah" after the completion of her iddah, in order to preserve her deceased husband's income, even though the prospective husband is married to a sick woman and will not inform her of his new marriage?

We ask Allah to have mercy on your sister's husband and to bless his family and offspring. Your intent by "Fatiha marriage" is not clear to us, but if the marriage fulfilled the conditions for validity, such as the guardian's permission and the presence of witnesses, then it is valid; otherwise, it is null and void. Concealing the marriage from some people does not harm it or make it a secret marriage according to the majority of scholars. The wealth left by her husband is an inheritance that all heirs are entitled to according to their prescribed share, based on the Prophet, peace and blessings be upon him, who said: "Whoever leaves behind wealth, it is for his heirs."