The Secretary shall pay to each child (1) who is the child of a deceased veteran of a period of war who met the service requirements prescribed in section 1521(j) of this title, or who at the time of death was receiving (or entitled to receive) compensation or retirement pay for a service-connected disability, and (2) who is not in the custody of a surviving spouse eligible for pension under section 1541 of this title, pension at the annual rate of $2,020, as increased from time to time under section 5312 of this title and reduced by the amount of such child’s annual income; or, if such child is residing with a person who is legally responsible for such child’s support, at an annual rate equal to the amount by which the appropriate annual rate provided under section 1541(c) of this title exceeds the sum of the annual income of such child and such person, but in no event may such annual rate of pension exceed the amount by which $2,020, as increased from time to time under section 5312 of this title, exceeds the annual income of such child. The appropriate annual rate under such section 1541(c) for the purposes of the preceding sentence shall be determined in accordance with regulations which the Secretary shall prescribe.