  In his study of finite (Vassiliev's) knot invariants,Vogel introduced the
so-called universal parameters, belonging to the projective plane, which
particularly parameterize the simple Lie algebras by the Vogel's table.
Subsequently a number of quantities, such as some universal knot invariants,
(quantum) dimensions of simple Lie algebras, etc., have been represented in
terms of these parameters, i.e. in the universal form. We prove that at the
points from the Vogel's table all known universal quantum dimension formulae
are linearly resolvable, i.e. yield finite answers even if these points are
singular, provided one restricts them to the appropriate lines. We show, that
the same phenomenon takes place for another three distinguished points in
Vogel's plane - $E_{7\frac{1}{2}}, X_1,$ and $X_2$. We also examine the same
formulae on linear resolvability at the remaining 48 distinguished points in
Vogel's plane, which correspond to the so-called $Y$-objects, and discover that
there are three points among them, which are regular for all known quantum
dimension formulae. Two of them happen to be sharing a remarkable similarity
with the simple Lie algebras, namely, the universal formulae yield
integer-valued outputs (dimensions) at those points in the classical limit.
