Create a new Dart file, let's call it supabase_service.dart.

Set up the Supabase client with your project's URL and API key:
```dart
import 'package:supabase/supabase.dart';

class SupabaseService {
final SupabaseClient client;

SupabaseService(String supabaseUrl, String supabaseKey)
: client = SupabaseClient(supabaseUrl, supabaseKey);
// Additional methods to interact with Supabase
}

final supabaseService = SupabaseService('YOUR_SUPABASE_URL', 'YOUR_SUPABASE_KEY');
```

Add methods for user authentication (sign-up and sign-in) in the supabase_service.dart file:
```dart
class SupabaseService {
final SupabaseClient client;

SupabaseService(String supabaseUrl, String supabaseKey)
: client = SupabaseClient(supabaseUrl, supabaseKey);
Future signUp(String email, String password) async {
final response = await client.auth.signUp(email, password);
return response;
}

Future signIn(String email, String password) async {
final response = await client.auth.signIn(email: email, password: password);
return response;
}
}

final supabaseService = SupabaseService('YOUR_SUPABASE_URL', 'YOUR_SUPABASE_KEY');
```

Create a simple Flutter UI to let users sign up and sign in:
```dart
import 'package:flutter/material.dart';

void main() {
runApp(MyApp());
}

class MyApp extends StatelessWidget {
@override
Widget build(BuildContext context) {
return MaterialApp(
title: 'Supabase with Flutter',
theme: ThemeData(
primarySwatch: Colors.blue,
),
home: SignUpPage(),
);
}
```

Create SignUpPage and SignInPage widgets and use the methods from the SupabaseService:

```dart
class SignUpPage extends StatefulWidget {
@override
_SignUpPageState createState() => _SignUpPageState();
}

class _SignUpPageState extends State {
final _emailController = TextEditingController();
final _passwordController = TextEditingController();

void _signUp() async {
final response = await supabaseService.signUp(
_emailController.text,
_passwordController.text,
);

if (response.error == null) {
// Handle user signed up successfully
} else {
// Handle error
}
}

@override
Widget build(BuildContext context) {
return Scaffold(
appBar: AppBar(
title: Text('Sign Up'),
),
body: Padding(
padding: const EdgeInsets.all(16.0),
child: Column(
children: [
TextField(
controller: _emailController,
decoration: InputDecoration(labelText: 'Email'),
),
TextField(
controller: _passwordController,
decoration: InputDecoration(labelText: 'Password'),
obscureText: true,
),
ElevatedButton(
onPressed: _signUp,
child: Text('Sign Up'),
),
TextButton(
onPressed: () {
Navigator.push(
context,
MaterialPageRoute(builder: (context) => SignInPage()),
);
},
child: Text('Already have an account? Sign In'),
),
],
),
),
);
}
}

class SignInPage extends StatefulWidget {
@override
_SignInPageState createState() => _SignInPageState();
}

class _SignInPageState extends State {
final _emailController = TextEditingController();
final _passwordController = TextEditingController();

void _signIn() async {
final response = await supabaseService.signIn(
_emailController.text,
_passwordController.text,
);

if (response.error == null) {
// Handle user signed in successfully
} else {
// Handle error
}
}

@override
Widget build(BuildContext context) {
return Scaffold(
appBar: AppBar(
title: Text('Sign In'),
),
body: Padding(
padding: const EdgeInsets.all(16.0),
child: Column(
children: [
TextField(
controller: _emailController,
decoration: InputDecoration(labelText: 'Email'),
),
TextField(
controller: _passwordController,
decoration: InputDecoration(labelText: 'Password'),
obscureText: true,
),
ElevatedButton(
onPressed: _signIn,
child: Text('Sign In'),
),
],
),
),
);
}
}
```

Make sure the Navigator is used effectively to switch between SignUpPage and SignInPage.