  Eigenproblems frequently arise in theory and applications of stochastic
processes, but only a few have explicit solutions. Those which do, are usually
solved by reduction to the generalized Sturm--Liouville theory for differential
operators. This includes the Brownian motion and a whole class of processes,
which derive from it by means of linear transformations. The more general
eigenproblem for the {\em fractional} Brownian motion (f.B.m.) is not solvable
in closed form, but the exact asymptotics of its eigenvalues and eigenfunctions
can be obtained, using a method based on analytic properties of the Laplace
transform. In this paper we consider two processes closely related to the
f.B.m.: the fractional Ornstein--Uhlenbeck process and the integrated
fractional Brownian motion. While both derive from the f.B.m. by simple linear
transformations, the corresponding eigenproblems turn out to be much more
complex and their asymptotic structure exhibits new effects.
