diff --git a/docs/src/examples/modelingtoolkitize_index_reduction.md b/docs/src/examples/modelingtoolkitize_index_reduction.md
index 8686fd60d4..b19ea46701 100644
--- a/docs/src/examples/modelingtoolkitize_index_reduction.md
+++ b/docs/src/examples/modelingtoolkitize_index_reduction.md
@@ -56,7 +56,7 @@ Specifically, for a pendulum with unit mass and length $L$, which thus has
 kinetic energy $\frac{1}{2}(v_x^2 + v_y^2)$,
 potential energy $gy$,
 and holonomic constraint $x^2 + y^2 - L^2 = 0$.
-The Lagrange multiplier related to this constraint is equal to half of $T$, 
+The Lagrange multiplier related to this constraint is equal to half of $T$,
 and represents the tension in the rope of the pendulum.
 
 As a good DifferentialEquations.jl user, one would follow
diff --git a/src/systems/systems.jl b/src/systems/systems.jl
index a54206d1dd..862718968d 100644
--- a/src/systems/systems.jl
+++ b/src/systems/systems.jl
@@ -154,8 +154,9 @@ function __structural_simplify(sys::AbstractSystem, io = nothing; simplify = fal
         end
 
         noise_eqs = StructuralTransformations.tearing_substitute_expr(ode_sys, noise_eqs)
-        return SDESystem(full_equations(ode_sys), noise_eqs,
+        return SDESystem(Vector{Equation}(full_equations(ode_sys)), noise_eqs,
             get_iv(ode_sys), unknowns(ode_sys), parameters(ode_sys);
-            name = nameof(ode_sys), is_scalar_noise, observed = observed(ode_sys))
+            name = nameof(ode_sys), is_scalar_noise, observed = observed(ode_sys), defaults = defaults(sys),
+            parameter_dependencies = parameter_dependencies(sys))
     end
 end
diff --git a/test/dde.jl b/test/dde.jl
index 2030a90d06..c7561e6c24 100644
--- a/test/dde.jl
+++ b/test/dde.jl
@@ -76,12 +76,13 @@ prob = SDDEProblem(hayes_modelf, hayes_modelg, [1.0], h, tspan, pmul;
     constant_lags = (pmul[1],));
 sol = solve(prob, RKMil(), seed = 100)
 
-@variables x(..)
+@variables x(..) delx(t)
 @parameters a=-4.0 b=-2.0 c=10.0 α=-1.3 β=-1.2 γ=1.1
 @brownian η
 τ = 1.0
-eqs = [D(x(t)) ~ a * x(t) + b * x(t - τ) + c + (α * x(t) + γ) * η]
+eqs = [D(x(t)) ~ a * x(t) + b * x(t - τ) + c + (α * x(t) + γ) * η, delx ~ x(t - τ)]
 @mtkbuild sys = System(eqs, t)
+@test ModelingToolkit.has_observed_with_lhs(sys, delx)
 @test ModelingToolkit.is_dde(sys)
 @test !is_markovian(sys)
 @test equations(sys) == [D(x(t)) ~ a * x(t) + b * x(t - τ) + c]