  The Constraint Satisfaction Problem (CSP) is a central and generic
computational problem which provides a common framework for many theoretical
and practical applications. A central line of research is concerned with the
identification of classes of instances for which CSP can be solved in
polynomial time; such classes are often called "islands of tractability." A
prominent way of defining islands of tractability for CSP is to restrict the
relations that may occur in the constraints to a fixed set, called a constraint
language, whereas a constraint language is conservative if it contains all
unary relations. This paper addresses the general limit of the mentioned
tractability results for CSP and #CSP, that they only apply to instances where
all constraints belong to a single tractable language (in general, the union of
two tractable languages isn't tractable). We show that we can overcome this
limitation as long as we keep some control of how constraints over the various
considered tractable languages interact with each other. For this purpose we
utilize the notion of a \emph{strong backdoor} of a CSP instance, as introduced
by Williams et al. (IJCAI 2003), which is a set of variables that when
instantiated moves the instance to an island of tractability, i.e., to a
tractable class of instances. In this paper, we consider strong backdoors into
\emph{scattered classes}, consisting of CSP instances where each connected
component belongs entirely to some class from a list of tractable classes. Our
main result is an algorithm that, given a CSP instance with $n$ variables,
finds in time $f(k)n^{O(1)}$ a strong backdoor into a scattered class
(associated with a list of finite conservative constraint languages) of size
$k$ or correctly decides that there isn't such a backdoor.
