mlxtend LDA vs sklearn LDA

Question

I'm trying to figure out 2 things regarding the LDA in mlxtend and the LDA on sklearn:

1. for 3 classes, why do the LDA results differ between mlxtend and sklearn
2. for 2 classes, why am I not able to get the same discriminants between mlxtend and sklearn.

everything is implemented on the iris dataset from mlxtend

from mlxtend.data import iris_data
from mlxtend.preprocessing import standardize
from mlxtend.feature_extraction import LinearDiscriminantAnalysis

X_iris, y_iris = iris_data()
X_iris = standardize(X_iris)

lda_iris = LinearDiscriminantAnalysis()
lda_iris.fit(X_iris, y_iris)
X_lda_iris = lda_iris.transform(X_iris)

from sklearn.discriminant_analysis import LinearDiscriminantAnalysis

clf = LinearDiscriminantAnalysis(solver='eigen')
clf.fit(X_iris, y_iris)
transformed_iris = clf.transform(X_iris)

import matplotlib.pyplot as plt

with plt.style.context('seaborn-whitegrid'):
    plt.figure(figsize=(6, 4))
    for lab, col in zip((0, 1, 2),
                        ('blue', 'red', 'green')):
        plt.scatter(X_lda_iris[y_iris == lab, 0],
                    X_lda_iris[y_iris == lab, 1],
                    label=lab,
                    c=col)
    plt.xlabel('Linear Discriminant 1')
    plt.ylabel('Linear Discriminant 2')
    plt.legend(loc='lower right')
    plt.tight_layout()
    plt.show()

import matplotlib.pyplot as plt

with plt.style.context('seaborn-whitegrid'):
    plt.figure(figsize=(6, 4))
    for lab, col in zip((0, 1, 2),
                        ('blue', 'red', 'green')):
        plt.scatter(transformed_iris[y_iris == lab, 0],
                    transformed_iris[y_iris == lab, 1],
                    label=lab,
                    c=col)
    plt.xlabel('Linear Discriminant 1')
    plt.ylabel('Linear Discriminant 2')
    plt.legend(loc='lower right')
    plt.tight_layout()
    plt.show()

you can see that the graphs are mirroring each other.

second thing: how come the LDA in mlxtend is able to return more than n-1 number of discriminants while sklearn returns n-1 (n= number of classes in the label).

I would expect both modules to give the same exact result.