  Linear spectral transformations of orthogonal polynomials in the real line,
and in particular Geronimus transformations, are extended to orthogonal
polynomials depending on several real variables. Multivariate
Christoffel-Geronimus-Uvarov formulae for the perturbed orthogonal polynomials
and their quasi-tau matrices are found for each perturbation of the original
linear functional. These expressions are given in terms of quasi-determinants
of bordered truncated block matrices and the 1D Christoffel-Geronimus-Uvarov
formulae in terms of quotient of determinants of combinations of the original
orthogonal polynomials and their Cauchy transforms, are recovered. A new
multispectral Toda hierarchy of nonlinear partial differential equations, for
which the multivariate orthogonal polynomials are reductions, is proposed. This
new integrable hierachy is associated with non-standard multivariate
biorthogonality. Wave and Baker functions, linear equations, Lax and
Zakharov-Shabat equations, KP type equations, appropriate reductions,
Darboux/linear spectral transformations, and bilinear equations involving
linear spectral transformations are presented.
