What was the background for ...?

"Private segregated academies" were schools founded for the express purpose of educationing white students only, and were supported strictly by private donations. They were part of an attempt to perpetuate racial segregation in the Southern section of the United States.

..."court-ordered integration of public schools". Since the states would not desegregate their schools, the Federal government stepped in.saying that the Constitution gave it the right to do so

",,,avoid the Constitutional and moral mandate". I don't believe the Constitution mentions education specifically in any Amendment, and arguments have been made that therefore desegregation is not mandatory. However, from various Amendments, denial of equal rights because of race, creed, color, or national origin has been legally interpreted and condemned as including such areas as education, thus overriding objections from the Southern states, and opening the way for desegregation of schools, public transportation, and any other public areas and services.

The Constitution states that any rights NOT specifically mentioned are reserved to the states, and it is this clause on which much of the argument in favor of desegration is based.

"The Civil Rights Act of 1964 (Pub.L. 88-352, 78 Stat. 241, July 2, 1964) was landmark legislation in the United States that outlawed segregation in the US schools and public places. First conceived to help African Americans, the bill was amended prior to passage to protect women in courts, and explicitly included white people for the first time. It also started the Equal Employment Opportunity Commission.

In order to circumvent limitations on the federal use of the Equal Protection Clause handed down by the Civil Rights Cases, the law was passed under the Commerce Clause. Once it was implemented, its effects were far reaching and had tremendous long-term impacts on the whole country. It prohibited discrimination in public facilities, in government, and in employment, invalidating the Jim Crow laws in the southern US. It became illegal to compel segregation of the races in schools, housing, or hiring. Powers given to enforce the bill were initially weak, but were supplemented during later years." - Wikipedia.

Our Constitution is narrow, in that it, and its Amendments, mention few detailed points, and broad in that it therefore leaves great room for interpretation as conditions in our country change throughout the years. It is this elasticity which is questioned by those Supreme Court Justices who would like literal, and therefore limiting, application; these Justices are of the "strict interpretationalist" school.