  The results from the article [Strachan I.A.B., Szablikowski B.M., Stud. Appl.
Math. 133 (2014), 84-117] are extended over consideration of central extensions
allowing the introducing of additional independent variables. Algebraic
conditions associated to the first-order central extension with respect to
additional independent variables are derived. As result $(2+1)$- and, in
principle, higher-dimensional multicomponent bi-Hamiltonian systems are
constructed. Necessary classification of the central extensions for
low-dimensional Novikov algebras is performed and the theory is illustrated by
significant $(2+1)$- and $(3+1)$-dimensional examples.
