  A constraint satisfaction problem (CSP) is a problem of computing a
homomorphism ${\bf R} \rightarrow {\bf \Gamma}$ between two relational
structures. Analyzing its complexity has been a very fruitful research
direction, especially for fixed template CSPs, denoted $CSP({\bf \Gamma})$, in
which the right side structure ${\bf \Gamma}$ is fixed and the left side
structure ${\bf R}$ is unconstrained.
  Recently, the hybrid setting, written $CSP_{\mathcal{H}}({\bf \Gamma})$,
where both sides are restricted simultaneously, attracted some attention. It
assumes that ${\bf R}$ is taken from a class of relational structures
$\mathcal{H}$ that additionally is closed under inverse homomorphisms. The last
property allows to exploit algebraic tools that have been developed for fixed
template CSPs. The key concept that connects hybrid CSPs with fixed-template
CSPs is the so called "lifted language". Namely, this is a constraint language
${\bf \Gamma}_{{\bf R}}$ that can be constructed from an input ${\bf R}$. The
tractability of that language for any input ${\bf R}\in\mathcal{H}$ is a
necessary condition for the tractability of the hybrid problem.
  In the first part we investigate templates ${\bf \Gamma}$ for which the
latter condition is not only necessary, but also is sufficient. We call such
templates ${\bf \Gamma}$ widely tractable. For this purpose, we construct from
${\bf \Gamma}$ a new finite relational structure ${\bf \Gamma}'$ and define
$\mathcal{H}_0$ as a class of structures homomorphic to ${\bf \Gamma}'$. We
prove that wide tractability is equivalent to the tractability of
$CSP_{\mathcal{H}_0}({\bf \Gamma})$. Our proof is based on the key observation
that ${\bf R}$ is homomorphic to ${\bf \Gamma}'$ if and only if the core of
${\bf \Gamma}_{{\bf R}}$ is preserved by a Siggers polymorphism. Analogous
result is shown for valued conservative CSPs.
