  The confluence of untyped lambda-calculus with unconditional rewriting has
already been studied in various directions. In this paper, we investigate the
confluence of lambda-calculus with conditional rewriting and provide general
results in two directions. First, when conditional rules are algebraic. This
extends results of Muller and Dougherty for unconditional rewriting. Two cases
are considered, whether beta-reduction is allowed or not in the evaluation of
conditions. Moreover, Dougherty's result is improved from the assumption of
strongly normalizing beta-reduction to weakly normalizing beta-reduction. We
also provide examples showing that outside these conditions, modularity of
confluence is difficult to achieve. Second, we go beyond the algebraic
framework and get new confluence results using an extended notion of
orthogonality that takes advantage of the conditional part of rewrite rules.
