A constraint satisfaction problem (CSP) is a problem of computing a homomorphism R -> G between two relational structures, e.g. between two directed graphs.

Analyzing its complexity has been a very fruitful research direction, especially for fixed template CSPs (or, non-uniform CSPs), denoted CSP(G),

in which the right side structure G is fixed and the left side structure R is unconstrained.

Recently, the hybrid setting, written CSP_H(G), where both sides are restricted simultaneously, attracted some attention.

It assumes that R is taken from a class of relational structures H (called the structural restriction) that additionally is closed under inverse homomorphisms. The last property allows to exploit an algebraic machinery that has 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 G_R that can be constructed from an input R. The tractability of the language G_R for any input R from H is a necessary condition for the tractability of the hybrid problem.

In the first part we investigate templates G for which the latter condition is not only necessary, but also is sufficient. We call such templates G widely tractable. For this purpose, we construct from G a new finite relational structure G' and define a maximal structural restriction H_0 as a class of structures homomorphic to G'.

For the so called strongly BJK templates that probably captures all templates, we prove that wide tractability is equivalent to the tractability of CSP_{H_0}(G).

Our proof is based on the key observation that R is homomorphic to G' if and only if the core of G_R is preserved by a Siggers polymorphism.

Analogous result is shown for conservative valued CSPs.