  We study two-leg S=1/2 ladders with general isotropic exchange interactions
between spins on neighboring rungs, whose ground state can be found exactly in
a form of finitely correlated (matrix product) wave function. Two families of
models admitting an exact solution are found: one yields translationally
invariant ground states and the other describes spontaneously dimerized models
with twofold degenerate ground state. Several known models with exact ground
states can be obtained as particular cases from the general solution of the
first family, which includes also a set of models with only bilinear
interactions. Those two families of models have nonzero intersection, which
enables us to determine exactly the phase boundary of the second-order
transition into the dimerized phase and to study the properties of this
transition. The structure of elementary excitations in the dimerized phase is
discussed on the basis of a variational ansatz. For a particular class of
models, we present exact wave functions of the elementary excitations becoming
gapless at second-order transition lines. We also propose a generalization of
the Bose-Gayen ladder model which has a rich phase diagram with all phase
boundaries being exact.
